Interesting Esoterica Thu, 30 Mar 2017 07:39:37 -0700 Approval Voting in Product Societies /entry/0 In approval voting, individuals vote for all platforms that they find acceptable. In this situation it is natural to ask: When is agreement possible? What conditions guarantee that some fraction of the voters agree on even a single platform? Berg et. al. found such conditions when voters are asked to make a decision on a single issue that can be represented on a linear spectrum. In particular, they showed that if two out of every three voters agree on a platform, there is a platform that is acceptable to a majority of the voters. Hardin developed an analogous result when the issue can be represented on a circular spectrum. We examine scenarios in which voters must make two decisions simultaneously. For example, if voters must decide on the day of the week to hold a meeting and the length of the meeting, then the space of possible options forms a cylindrical spectrum. Previous results do not apply to these multi-dimensional voting societies because a voter's preference on one issue often impacts their preference on another. We present a general lower bound on agreement in a two-dimensional voting society, and then examine specific results for societies whose spectra are cylinders and tori. Wed, 31 Dec 1969 16:33:37 -0800 Kristen Mazur and Mutiara Sondjaja and Matthew Wright and Carolyn Yarnall Pauli Pascal Pyramids, Pauli Fibonacci Numbers, and Pauli Jacobsthal Numbers /entry/1 The three anti-commutative two-dimensional Pauli Pascal triangles can be generalized into multi-dimensional Pauli Pascal hyperpyramids. Fibonacci and Jacobsthal numbers are then generalized into Pauli Fibonacci numbers, Pauli Jacobsthal numbers, and Pauli Fibonacci numbers of higher order. And the question is: are Pauli rabbits killer rabbits? 1 Wed, 31 Dec 1969 16:33:37 -0800 Martin Erik Horn Jewish Problems /entry/2 This is a special collection of problems that were given to select applicants during oral entrance exams to the math department of Moscow State University. These problems were designed to prevent Jews and other undesirables from getting a passing grade. Among problems that were used by the department to blackball unwanted candidate students, these problems are distinguished by having a simple solution that is difficult to find. Using problems with a simple solution protected the administration from extra complaints and appeals. This collection therefore has mathematical as well as historical value. 2 Wed, 31 Dec 1969 16:33:37 -0800 Tanya Khovanova and Alexey Radul Best Laid Plans of Lions and Men /entry/3 We answer the following question dating back to J.E. Littlewood (1885 - 1977): Can two lions catch a man in a bounded area with rectifiable lakes? The lions and the man are all assumed to be points moving with at most unit speed. That the lakes are rectifiable means that their boundaries are finitely long. This requirement is to avoid pathological examples where the man survives forever because any path to the lions is infinitely long. We show that the answer to the question is not always "yes" by giving an example of a region $R$ in the plane where the man has a strategy to survive forever. $R$ is a polygonal region with holes and the exterior and interior boundaries are pairwise disjoint, simple polygons. Our construction is the first truly two-dimensional example where the man can survive. Next, we consider the following game played on the entire plane instead of a bounded area: There is any finite number of unit speed lions and one fast man who can run with speed $1+\varepsilon$ for some value $\varepsilon>0$. Can the man always survive? We answer the question in the affirmative for any constant $\varepsilon>0$. 3 Wed, 31 Dec 1969 16:33:37 -0800 Mikkel Abrahamsen and Jacob Holm and Eva Rotenberg and Christian Wulff-Nilsen PAPAC-00, a Do-It-Yourself Paper Computer /entry/4 4 Wed, 31 Dec 1969 16:33:37 -0800 Rollin P. Mayer Beyond Floating Point: Next-Generation Computer Arithmetic /entry/5 5 Wed, 31 Dec 1969 16:33:37 -0800 John L. Gustafson Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9 /entry/6 Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two different ways. The first one in increasing order of 1 to 9, and the second one in decreasing order. This is done by using the operations of addition, multiplication, subtraction, potentiation, and division. In both the situations there are no missing numbers, except one, i.e., 10958 in the increasing case. 6 Wed, 31 Dec 1969 16:33:37 -0800 Inder J. Taneja The mathematics of lecture hall partitions /entry/7 Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to lecture hall partitions have used polyhedral geometry to discover further properties of these rich combinatorial objects. In this paper we give an overview of some of the surprising connections that have surfaced in the process of trying to understand the lecture hall partitions. 7 Wed, 31 Dec 1969 16:33:37 -0800 Carla D. Savage Statistics Done Wrong /entry/8 If you’re a practicing scientist, you probably use statistics to analyze your data. From basic t tests and standard error calculations to Cox proportional hazards models and propensity score matching, we rely on statistics to give answers to scientific problems. This is unfortunate, because statistical errors are rife. Statistics Done Wrong is a guide to the most popular statistical errors and slip-ups committed by scientists every day, in the lab and in peer-reviewed journals. Many of the errors are prevalent in vast swaths of the published literature, casting doubt on the findings of thousands of papers. Statistics Done Wrong assumes no prior knowledge of statistics, so you can read it before your first statistics course or after thirty years of scientific practice. 8 Wed, 31 Dec 1969 16:33:37 -0800 Alex Reinhart Prime Simplicity /entry/9 9 Wed, 31 Dec 1969 16:33:37 -0800 Michael Hardy and Catherine Woodgold Meaning in Classical Mathematics: Is it at Odds with Intuitionism? /entry/10 We examine the classical/intuitionist divide, and how it reflects on modern theories of infinitesimals. When leading intuitionist Heyting announced that "the creation of non-standard analysis is a standard model of important mathematical research", he was fully aware that he was breaking ranks with Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a comparative textual analysis of three of Bishop's texts, we analyze the ideological and/or pedagogical nature of his objections to infinitesimals a la Robinson. Bishop's famous "debasement" comment at the 1974 Boston workshop, published as part of his Crisis lecture, in reality was never uttered in front of an audience. We compare the realist and the anti-realist intuitionist narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and Tennant. Variational principles are important physical applications, currently lacking a constructive framework. We examine the case of the Hawking-Penrose singularity theorem, already analyzed by Hellman in the context of the Quine-Putnam indispensability thesis. 10 Wed, 31 Dec 1969 16:33:37 -0800 Karin Usadi Katz and Mikhail G. Katz Three Thoughts on “Prime Simplicity” /entry/11 In 2009, Catherine Woodgold and I published ‘‘Prime Simplicity’’, examining the belief that Euclid’s famous proof of the infinitude of prime numbers was by contradiction. We demonstrated that that belief is widespread among mathematicians and is false: Euclid’s proof is simpler and better than the frequently seen proof by contradiction. The extra complication of the indirect proof serves no purpose and has pitfalls that can mislead the reader. 11 Wed, 31 Dec 1969 16:33:37 -0800 Michael Hardy Plane partitions in the work of Richard Stanley and his school /entry/12 These notes provide a survey of the theory of plane partitions, seen through the glasses of the work of Richard Stanley and his school. 12 Wed, 31 Dec 1969 16:33:37 -0800 C. Krattenthaler On the Existence of Ordinary Triangles /entry/13 Let $P$ be a finite point set in the plane. A $c$-ordinary triangle in $P$ is a subset of $P$ consisting of three non-collinear points such that each of the three lines determined by the three points contains at most $c$ points of $P$. We prove that there exists a constant $c>0$ such that $P$ contains a $c$-ordinary triangle, provided that $P$ is not contained in the union of two lines. Furthermore, the number of $c$-ordinary triangles in $P$ is $\Omega(|P|)$. 13 Wed, 31 Dec 1969 16:33:37 -0800 Radoslav Fulek and Hossein Nassajian Mojarrad and Márton Naszódi and József Solymosi and Sebastian U. Stich and May Szedlák Random Triangles and Polygons in the Plane /entry/14 We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. In these terms, we answer Caroll's question. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space of unordered quadrilaterals. All of this provides a concrete introduction to a family of metrics used in shape classification and computer vision. 14 Wed, 31 Dec 1969 16:33:37 -0800 Jason Cantarella and Tom Needham and Clayton Shonkwiler and Gavin Stewart Analysis of Carries in Signed Digit Expansions /entry/15 The number of positive and negative carries in the addition of two independent random signed digit expansions of given length is analyzed asymptotically for the $(q, d)$-system and the symmetric signed digit expansion. The results include expectation, variance, covariance between the positive and negative carries and a central limit theorem. Dependencies between the digits require determining suitable transition probabilities to obtain equidistribution on all expansions of given length. A general procedure is described to obtain such transition probabilities for arbitrary regular languages. The number of iterations in von Neumann's parallel addition method for the symmetric signed digit expansion is also analyzed, again including expectation, variance and convergence to a double exponential limiting distribution. This analysis is carried out in a general framework for sequences of generating functions. 15 Wed, 31 Dec 1969 16:33:37 -0800 Clemens Heuberger and Sara Kropf and Helmut Prodinger Transfinite Version of Welter's Game /entry/16 We study the transfinite version of Welter's Game, a combinatorial game, which is played on the belt divided into squares with general ordinal numbers extended from natural numbers. In particular, we obtain a straight-forward solution for the transfinite version based on those of the transfinite version of Nim and the original version of Welter's Game. 16 Wed, 31 Dec 1969 16:33:37 -0800 Tomoaki Abuku Hunting Rabbits on the Hypercube /entry/17 We explore the Hunters and Rabbits game on the hypercube. In the process, we find the solution for all classes of graphs with an isoperimetric nesting property and find the exact hunter number of $Q^n$ to be $1+\sum\limits_{i=0}^{n-2} \binom{i}{\lfloor i/2 \rfloor}$. In addition, we extend results to the situation where we allow the rabbit to not move between shots. 17 Wed, 31 Dec 1969 16:33:37 -0800 Jessalyn Bolkema and Corbin Groothuis Rules for Folding Polyminoes from One Level to Two Levels /entry/18 Polyominoes have been the focus of many recreational and research investigations. In this article, the authors investigate whether a paper cutout of a polyomino can be folded to produce a second polyomino in the same shape as the original, but now with two layers of paper. For the folding, only "corner folds" and "half edge cuts" are allowed, unless the polyomino forms a closed loop, in which case one is allowed to completely cut two squares in the polyomino apart. With this set of allowable moves, the authors present algorithms for folding different types of polyominoes and prove that certain polyominoes can successfully be folded to two layers. The authors also establish that other polyominoes cannot be folded to two layers if only these moves are allowed. 18 Wed, 31 Dec 1969 16:33:37 -0800 Julia Martin and Elizabeth Wilcox Human Inferences about Sequences: A Minimal Transition Probability Model /entry/19 The brain constantly infers the causes of the inputs it receives and uses these inferences to generate statistical expectations about future observations. Experimental evidence for these expectations and their violations include explicit reports, sequential effects on reaction times, and mismatch or surprise signals recorded in electrophysiology and functional MRI. Here, we explore the hypothesis that the brain acts as a near-optimal inference device that constantly attempts to infer the time-varying matrix of transition probabilities between the stimuli it receives, even when those stimuli are in fact fully unpredictable. This parsimonious Bayesian model, with a single free parameter, accounts for a broad range of findings on surprise signals, sequential effects and the perception of randomness. Notably, it explains the pervasive asymmetry between repetitions and alternations encountered in those studies. Our analysis suggests that a neural machinery for inferring transition probabilities lies at the core of human sequence knowledge. 19 Wed, 31 Dec 1969 16:33:37 -0800 Florent Meyniel and Maxime Maheu and Stanislas Dehaene A Singular Mathematical Promenade /entry/20 This is neither an elementary introduction to singularity theory nor a specialized treatise containing many new theorems. The purpose of this little book is to invite the reader on a mathematical promenade. We will pay a visit to Hipparchus, Newton and Gauss, but also to many contemporary mathematicians. We will play with a bit of algebra, topology, geometry, complex analysis and computer science. Hopefully, some motivated undergraduates and some more advanced mathematicians will enjoy some of these panoramas. 20 Wed, 31 Dec 1969 16:33:36 -0800 Etienne Ghys Balloon Polyhedra /entry/21 21 Wed, 31 Dec 1969 16:33:36 -0800 Erik D. Demaine and Martin L. Demaine and Vi Hart Two short proofs of the Perfect Forest Theorem /entry/22 A perfect forest is a spanning forest of a connected graph $G$, all of whose components are induced subgraphs of $G$ and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a matching, all components are trees on one edge. Scott first proved the Perfect Forest Theorem, namely, that every connected graph of even order has a perfect forest. Gutin then gave another proof using linear algebra. We give here two very short proofs of the Perfect Forest Theorem which use only elementary notions from graph theory. Both our proofs yield polynomial-time algorithms for finding a perfect forest in a connected graph of even order. 22 Wed, 31 Dec 1969 16:33:36 -0800 Yair Caro and Josef Lauri and Christina Zarb Every natural number is the sum of forty-nine palindromes /entry/23 It is shown that the set of decimal palindromes is an additive basis for the natural numbers. Specifically, we prove that every natural number can be expressed as the sum of forty-nine (possibly zero) decimal palindromes. 23 Wed, 31 Dec 1969 16:33:36 -0800 William D. Banks Sequences of consecutive \(n\)-Niven numbers /entry/24 A Niven number is a positive integer that is divisible by the sum of its digits. In 1982, Kennedy showed that there do not exist sequences of more than 21 consecutive Niven numbers. In 1992, Cooper & Kennedy improved this result by proving that there does not exist a sequence of more than 20 consecutive Niven numbers. They also proved that this bound is the best possible by producing an infinite family of sequences of 20 consecutive Niven numbers. For any positive integer \(n \gt 2\), define an \(n\)-Niven number to be a positive integer that is divisible by the sum of the digits in its base \(n\) expansion. This paper examines the maximal possible lengths of sequences of consecutive \(n\)-Niven numbers. The main result is given in the following theorem. 24 Wed, 31 Dec 1969 16:33:36 -0800 H.G. Grundman Developing a Mathematical Model for Bobbin Lace /entry/25 Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. An overview of how bobbin lace is made is presented and illustrated with a simple, traditional bookmark design. Research on the topology of textiles and braid theory form a base for the current work and is briefly summarized. We define a new mathematical model that supports the enumeration and generation of bobbin lace patterns using an intelligent combinatorial search. Results of this new approach are presented and, by comparison to existing bobbin lace patterns, it is demonstrated that this model reveals new patterns that have never been seen before. Finally, we apply our new patterns to an original bookmark design and propose future areas for exploration. 25 Wed, 31 Dec 1969 16:33:36 -0800 Veronika Irvine and Frank Ruskey Quasipractical Numbers /entry/26 26 Wed, 31 Dec 1969 16:33:36 -0800 Harvey J. Hindin Cryptographic Protocols with Everyday Objects /entry/27 Most security protocols appearing in the literature make use of cryptographic primitives that assume that the participants have access to some sort of computational device. However, there are times when there is need for a security mechanism to evaluate some result without leaking sensitive information, but computational devices are unavailable. We discuss here various protocols for solving cryptographic problems using everyday objects: coins, dice, cards, and envelopes. 27 Wed, 31 Dec 1969 16:33:36 -0800 James Heather and Steve Schneider and Vanessa Teague On the interval containing at least one prime number /entry/28 28 Wed, 31 Dec 1969 16:33:36 -0800 Jitsuro Nagura On subsets with intersections of even cardinality /entry/29 This paper solves a question by Paul Erdős 29 Wed, 31 Dec 1969 16:33:36 -0800 E.R. Berlekamp Two remarks on even and oddtown problems /entry/30 A family $\mathcal A$ of subsets of an $n$-element set is called an eventown (resp. oddtown) if all its sets have even (resp. odd) size and all pairwise intersections have even size. Using tools from linear algebra, it was shown by Berlekamp and Graver that the maximum size of an eventown is $2^{\left\lfloor n/2\right\rfloor}$. On the other hand (somewhat surprisingly), it was proven by Berlekamp, that oddtowns have size at most $n$. Over the last four decades, many extensions of this even/oddtown problem have been studied. In this paper we present new results on two such extensions. First, extending a result of Vu, we show that a $k$-wise eventown (i.e., intersections of $k$ sets are even) has for $k \geq 3$ a unique extremal configuration and obtain a stability result for this problem. Next we improve some known bounds for the defect version of an $\ell$-oddtown problem. In this problem we consider sets of size $\not\equiv 0 \pmod \ell$ where $\ell$ is a prime number $\ell$ (not necessarily $2$) and allow a few pairwise intersections to also have size $\not\equiv 0 \pmod \ell$. 30 Wed, 31 Dec 1969 16:33:36 -0800 Benny Sudakov and Pedro Vieira A Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence /entry/31 With standard algorithms for generating the classical Kolakoski sequence, the numerical calculation of the digit distribution requires a linear amount of space. Here, we present an algorithm for calculating the distribution of the digits in the classical Kolakoski sequence, that only requires a logarithmic amount of space and still runs in linear time. The algorithm is easily adaptable to generalised Kolakoski sequences. 31 Wed, 31 Dec 1969 16:33:36 -0800 Johan Nilsson Geometric Mechanics of Curved Crease Origami /entry/32 Folding a sheet of paper along a curve can lead to structures seen in decorative art and utilitarian packing boxes. Here we present a theory for the simplest such structure: an annular circular strip that is folded along a central circular curve to form a three-dimensional buckled structure driven by geometrical frustration. We quantify this shape in terms of the radius of the circle, the dihedral angle of the fold and the mechanical properties of the sheet of paper and the fold itself. When the sheet is isometrically deformed everywhere except along the fold itself, stiff folds result in creases with constant curvature and oscillatory torsion. However, relatively softer folds inherit the broken symmetry of the buckled shape with oscillatory curvature and torsion. Our asymptotic analysis of the isometrically deformed state is corroborated by numerical simulations which allow us to generalize our analysis to study multiply folded structures. 32 Wed, 31 Dec 1969 16:33:36 -0800 Marcelo A. Dias and Levi H. Dudte and L. Mahadevan and Christian D. Santangelo A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents /entry/33 We consider the well-studied cake cutting problem in which the goal is to find an envy-free allocation based on queries from $n$ agents. The problem has received attention in computer science, mathematics, and economics. It has been a major open problem whether there exists a discrete and bounded envy-free protocol. We resolve the problem by proposing a discrete and bounded envy-free protocol for any number of agents. The maximum number of queries required by the protocol is $n^{n^{n^{n^{n^n}}}}$. We additionally show that even if we do not run our protocol to completion, it can find in at most $n^{n+1}$ queries a partial allocation of the cake that achieves proportionality (each agent gets at least $1/n$ of the value of the whole cake) and envy-freeness. Finally we show that an envy-free partial allocation can be computed in $n^{n+1}$ queries such that each agent gets a connected piece that gives the agent at least $1/(3n)$ of the value of the whole cake. 33 Wed, 31 Dec 1969 16:33:36 -0800 Haris Aziz and Simon Mackenzie Avoiding Squares and Overlaps Over the Natural Numbers /entry/34 We consider avoiding squares and overlaps over the natural numbers, using a greedy algorithm that chooses the least possible integer at each step; the word generated is lexicographically least among all such infinite words. In the case of avoiding squares, the word is 01020103..., the familiar ruler function, and is generated by iterating a uniform morphism. The case of overlaps is more challenging. We give an explicitly-defined morphism phi : N* -> N* that generates the lexicographically least infinite overlap-free word by iteration. Furthermore, we show that for all h,k in N with h <= k, the word phi^{k-h}(h) is the lexicographically least overlap-free word starting with the letter h and ending with the letter k, and give some of its symmetry properties. 34 Wed, 31 Dec 1969 16:33:36 -0800 Mathieu Guay-Paquet and Jeffrey Shallit Counting Cases in Marching Cubes: Towards a Generic Algorithm for Producing Substitopes /entry/35 We describe how to count the cases that arise in a family of visualization techniques, including marching cubes, sweeping simplices, contour meshing, interval volumes, and separating surfaces. Counting the cases is the first step toward developing a generic visualization algorithm to produce substitopes (geometric substitution of polytopes). We demonstrate the method using a software system ("GAP") for computational group theory. The case-counts are organized into a table that provides taxonomy of members of the family; numbers in the table are derived from actual lists of cases, which are computed by our methods. The calculation confirms previously reported case-counts for large dimensions that are too large to check by hand, and predicts the number of cases that will arise in algorithms that have not yet been invented. 35 Wed, 31 Dec 1969 16:33:36 -0800 David C. Banks and Stephen Linton Fractal geometry of a complex plumage trait reveals bird's quality /entry/36 Animal coloration is key in natural and sexual selection, playing significant roles in intra- and interspecific communication because of its linkage to individual behaviour, genetics and physiology. Simple animal traits such as the area or the colour intensity of homogeneous patches have been profusely studied. More complex patterns are widespread in nature, but they escape our understanding because their variation is difficult to capture effectively by standard, simple measures. Here, we used fractal geometry to quantify inter-individual variation in the expression of a complex plumage trait, the heterogeneous black bib of the red-legged partridge (Alectoris rufa). We show that a higher bib fractal dimension (FD) predicted better individual body condition, as well as immune responsiveness, which is condition-dependent in our study species. Moreover, when food intake was experimentally reduced during moult as a means to reduce body condition, the bib's FD significantly decreased. Fractal geometry therefore provides new opportunities for the study of complex animal colour patterns and their roles in animal communication. 36 Wed, 31 Dec 1969 16:33:36 -0800 Lorenzo Pérez-Rodríguez and Roger Jovani and Fran\ccois Mougeot Programming quantum computers using 3-D puzzles, coffee cups, and doughnuts /entry/37 The task of programming a quantum computer is just as strange as quantum mechanics itself. But it now looks like a simple 3D puzzle may be the future tool of quantum software engineers. 37 Wed, 31 Dec 1969 16:33:36 -0800 Simon J. Devitt The Nesting and Roosting Habits of The Laddered Parenthesis /entry/38 38 Wed, 31 Dec 1969 16:33:36 -0800 R. K. Guy and J. L. Selfridge Historical methods for multiplication /entry/39 This paper summarizes the contents of our workshop. In this workshop, we presented and discussed the "Greek" multiplication, given by Eutokios of Ascalon in his commentary on The Measurement of a Circle. We discussed part of the text from the treatise of Eutokios. Our basic thesis is that we think that this historical method for multiplication is part of the algorithms friendly to the user (based on the ideas that the children use in their informal mental strategies). The important idea is that the place value of numbers is maintained and the students act with quantities and not with isolated symbols as it happens with the classic algorithm. This helps students to control their thought at every stage of calculation. We also discussed the Russian method and the method by the cross (basically the same as "Casting out nines") to control the execution of the operations. 39 Wed, 31 Dec 1969 16:33:36 -0800 Bjørn Smestad and Konstantinos Nikolantonakis Ponytail Motion /entry/40 A jogger's ponytail sways from side to side as the jogger runs, although her head does not move from side to side. The jogger's head just moves up and down, forcing the ponytail to do so also. We show in two ways that this vertical motion is unstable to lateral perturbations. First we treat the ponytail as a rigid pendulum, and then we treat it as a flexible string; in each case, it is hanging from a support which is moving up and down periodically, and we solve the linear equation for small lateral oscillation. The angular displacement of the pendulum and the amplitude of each mode of the string satisfy Hill's equation. This equation has solutions which grow exponentially in time when the natural frequency of the pendulum, or that of a mode of the string, is close to an integer multiple of half the frequency of oscillation of the support. Then the vertical motion is unstable, and the ponytail sways. 40 Wed, 31 Dec 1969 16:33:36 -0800 Joseph B. Keller Seven Puzzles You Think You Must Not Have Heard Correctly /entry/41 A typical mathematical puzzle sounds tricky but solvable — if not by you, then perhaps by the genius down the hall. But sometimes the task at hand is so obviously impossible that you are moved to ask whether you understood the problem correctly, and other times, the task seems so trivial that you are sure you must have missed something. Here, I have compiled seven puzzles which have often been greeted by words similar to “Wait a minute — I must not have heard that correctly.” Some seem too hard, some too easy; after you've worked on them for a while, you may find that the hard ones now seem easy and vice versa. 41 Wed, 31 Dec 1969 16:33:36 -0800 Peter Winkler Topologically Distinct Sets of Non-intersecting Circles in the Plane /entry/42 Nested parentheses are forms in an algebra which define orders of evaluations. A class of well-formed sets of associated opening and closing parentheses is well studied in conjunction with Dyck paths and Catalan numbers. Nested parentheses also represent cuts through circles on a line. These become topologies of non-intersecting circles in the plane if the underlying algebra is commutative. This paper generalizes the concept and answers quantitatively - as recurrences and generating functions of matching rooted forests - the questions: how many different topologies of nested circles exist in the plane if (i) pairs of circles may intersect, or (ii) even triples of circles may intersect. That analysis is driven by examining the symmetry properties of the inner regions of the fundamental type(s) of the intersecting pairs and triples. 42 Wed, 31 Dec 1969 16:33:36 -0800 Richard J. Mathar Beckett-Gray Codes /entry/43 In this paper we discuss a natural mathematical structure that is derived from Samuel Beckett's play "Quad". This structure is called a binary Beckett-Gray code. Our goal is to formalize the definition of a binary Beckett-Gray code and to present the work done to date. In addition, we describe the methodology used to obtain enumeration results for binary Beckett-Gray codes of order $n = 6$ and existence results for binary Beckett-Gray codes of orders $n = 7,8$. We include an estimate, using Knuth's method, for the size of the exhaustive search tree for $n=7$. Beckett-Gray codes can be realized as successive states of a queue data structure. We show that the binary reflected Gray code can be realized as successive states of two stack data structures. 43 Wed, 31 Dec 1969 16:33:36 -0800 Mark Cooke and Chris North and Megan Dewar and Brett Stevens The general counterfeit coin problem /entry/44 Given $c$ nickels among which there may be a counterfeit coin, which can only be told apart by its weight being different from the others, and moreover $b$ balances. What is the minimal number of weighings to decide whether there is a counterfeit nickel, if so which one it is and whether it is heavier or lighter than a genuine nickel. We give an answer to this question for sequential and nonsequential strategies and we will consider the problem of more than one counterfeit coin. 44 Wed, 31 Dec 1969 16:33:36 -0800 Lorenz Halbeisen and Norbert Hungerbühler Searching for generalized binary number systems /entry/45 The aim of the project is to find all the generalized binary number systems up to dimension 11. Below we give a short description of the number system concept and mention a few possible applications. 45 Wed, 31 Dec 1969 16:33:36 -0800 Attila Kovács The denominators of convergents for continued fractions /entry/46 For any real number $x \in [0,1)$, we denote by $q_n(x)$ the denominator of the $n$-th convergent of the continued fraction expansion of $x$ $(n \in \mathbb{N})$. It is well-known that the Lebesgue measure of the set of points $x \in [0,1)$ for which $\log q_n(x)/n$ deviates away from $\pi^2/(12\log2)$ decays to zero as $n$ tends to infinity. In this paper, we study the rate of this decay by giving an upper bound and a lower bound. What is interesting is that the upper bound is closely related to the Hausdorff dimensions of the level sets for $\log q_n(x)/n$. As a consequence, we obtain a large deviation type result for $\log q_n(x)/n$, which indicates that the rate of this decay is exponential. 46 Wed, 31 Dec 1969 16:33:36 -0800 Lulu Fang and Min Wu and Bing Li Matters Computational - Ideas, Algorithms, Source Code /entry/47 This is the book "Matters Computational" (formerly titled "Algorithms for Programmers"), published with Springer. 47 Wed, 31 Dec 1969 16:33:36 -0800 Jörg Arndt Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations /entry/48 48 Wed, 31 Dec 1969 16:33:36 -0800 Giancarlo Rota Rational Polynomials That Take Integer Values at the Fibonacci Numbers /entry/49 An integer-valued polynomial on a subset $S$ of $\mathbb{Z}$ is a polynomial $f(x) \in \mathbb{Q}[x]$ with the property $f(S) \subseteq \mathbb{Z}$. This article describes the ring of such polynomials in the special case that $S$ is the Fibonacci numbers. An algorithm is described for finding a regular basis, i.e., an ordered sequence of polynomials, the $n$th one of degree $n$, with which any such polynomial can be expressed as a unique integer linear combination. 49 Wed, 31 Dec 1969 16:33:36 -0800 Keith Johnson and Kira Scheibelhut Bad groups in the sense of Cherlin /entry/50 There exists no bad group (in the sense of Gregory Cherlin), namely any simple group of Morley rank 3 is isomorphic to $\mathrm{PSL_2}(K)$ for an algebraically closed field $K$. 50 Wed, 31 Dec 1969 16:33:36 -0800 Olivier Frécon An Irrationality Measure for Regular Paperfolding Numbers /entry/51 Let $F(z) = \sum_{n \geq 1} f_n z^n$ be the generating series of the regular paperfolding sequence. For a real number $\alpha$ the irrationality exponent $\mu(\alpha)$, of $\alpha$, is defined as the supremum of the set of real numbers $\mu$ such that the inequality $\lvert \alpha - p/q \rvert \lt q-\mu$ has infinitely many solutions $(p,q) \in Z \times N$. In this paper, using a method introduced by Bugeaud, we prove that \[ \mu(F(1/b)) \leq 275331112987/137522851840 = 2.002075359 \ldots \] for all integers $b \geq 2$. This improves upon the previous bound of $\mu(F(1/b)) \leq 5$ given by Adamczewski and Rivoal. 51 Wed, 31 Dec 1969 16:33:36 -0800 Michael Coons and Paul Vrbik The nesting and roosting habits of the laddered parenthesis /entry/52 52 Wed, 31 Dec 1969 16:33:36 -0800 R.K. Guy and J. L. Selfridge What is the smallest prime? /entry/53 What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers. To find the first prime, we must also know what the first positive integer is. Surprisingly, with the definitions used at various times throughout history, one was often not the first positive integer (some started with two, and a few with three). In this article, we survey the history of the primality of one, from the ancient Greeks to modern times. We will discuss some of the reasons definitions changed, and provide several examples. We will also discuss the last significant mathematicians to list the number one as prime. 53 Wed, 31 Dec 1969 16:33:36 -0800 Chris K. Caldwell and Yeng Xiong How do you compute the midpoint of an interval? /entry/54 54 Wed, 31 Dec 1969 16:33:36 -0800 Frédéric Goualard Complexity and Completeness of Finding Another solution and its Application to Puzzles /entry/55 The Another Solution Problem (ASP) of a problem $\Pi$ is the following problem: for a given instance $x$ of $\Pi$ and a solution $s$ to it, find a solution to $x$ other than $s$. The notion of ASP as a new class of problems was first introduced by Ueda and Nagao. They also pointed out that polynomial-time parsimonious reductions which allow polynomial-time transformation of solutions can derive the NP-completeness of ASP of a certain problem from that of ASP of another. They used this property to show the NP-completeness of ASP of Nonogram, a sort of puzzle. Following it, Seta considered the problem to find another solution when $n$ solutions are given. (We call the problem $n$-ASP.) He proved the NP-completeness of $n$-ASP of some problems, including Cross Sum, for any $n$. In this thesis we establish a rigid formalization of $n$-ASPs to investigate their characteristics more clearly. In particular we introduce ASP-completeness, the completeness with respect to the reductions satisfying the properties mentioned above, and show that ASP-completeness of a problem implies NP-completeness of $n$-ASP of the problem for all $n$. Moreover we research the relation between ASPs and other versions of problems, such as counting problems and enumeration problems, and show the equivalence of the class of problems which allow enumerations of solutions in polynomial time and the class of problems of which $n$-ASP is solvable in polynomial time. As Ueda and Nagao pointed out, the complexity of ASPs has a relation with the difficulty of designing puzzles. We prove the ASP-completeness of three popular puzzles: Slither Link, Number Place and Fillomino. The ASP-completeness of Slither Link is shown via a reduction from the Hamiltonian circuit problem for restricted graphs, that of Number Place is from the problem of Latin square completion, and that of Fillomino is from planar 3SAT. Since ASP=completeness implies NP-completeness as is mentioned above, these results can be regarded as new results of NP-completeness proof of puzzles. 55 Wed, 31 Dec 1969 16:33:36 -0800 Takayushi Yato Dividing by zero - how bad is it, really? /entry/56 In computable analysis testing a real number for being zero is a fundamental example of a non-computable task. This causes problems for division: We cannot ensure that the number we want to divide by is not zero. In many cases, any real number would be an acceptable outcome if the divisor is zero - but even this cannot be done in a computable way. In this note we investigate the strength of the computational problem "Robust division": Given a pair of real numbers, the first not greater than the other, output their quotient if well-defined and any real number else. The formal framework is provided by Weihrauch reducibility. One particular result is that having later calls to the problem depending on the outcomes of earlier ones is strictly more powerful than performing all calls concurrently. However, having a nesting depths of two already provides the full power. This solves an open problem raised at a recent Dagstuhl meeting on Weihrauch reducibility. As application for "Robust division", we show that it suffices to execute Gaussian elimination. 56 Wed, 31 Dec 1969 16:33:36 -0800 Takayuki Kihara and Arno Pauly Fuzzy plane geometry I: Points and lines /entry/57 We introduce a comprehensive study of fuzzy geometry in this paper by first defining a fuzzy point and a fuzzy line in fuzzy plane geometry. We consider the fuzzy distance between fuzzy points and show it is a (weak) fuzzy metric. We study various definitions of a fuzzy line, develop their basic properties, and investigate parallel fuzzy lines. 57 Wed, 31 Dec 1969 16:33:36 -0800 J.J. Buckley and E. Aslami Continued Logarithms And Associated Continued Fractions /entry/58 We investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the continued logarithm to base $b$. We show convergence for them using equivalent forms of their corresponding continued fractions. Through numerical experimentation we discover that, for one such formulation, the exponent terms have finite arithmetic means for almost all real numbers. This set of means, which we call the logarithmic Khintchine numbers, has a pleasing relationship with the geometric means of the corresponding continued fraction terms. While the classical Khintchine’s constant is believed not to be related to any naturally occurring number, we find surprisingly that the logarithmic Khintchine numbers are elementary. 58 Wed, 31 Dec 1969 16:33:36 -0800 Jonathan M. Borwein and Neil J. Calkin and Scott B. Lindstrom and Andrew Mattingly Notes on the Fourth Dimension /entry/59 Hyperspace, ghosts, and colourful cubes — Jon Crabb on the work of Charles Howard Hinton and the cultural history of higher dimensions. 59 Wed, 31 Dec 1969 16:33:36 -0800 Jon Crabb Dr Mitchill and the Mathematical Tetrodon /entry/60 60 Wed, 31 Dec 1969 16:33:36 -0800 Kevin Dann Photoelectric Number Sieve Machine ("Gear Machine") /entry/61 This gear number sieve was constructed to solve number theory problems such as factoring and determining if a number is prime. The machine also had a photo detector and powerful amplifier which was not included in the gift. 61 Wed, 31 Dec 1969 16:33:36 -0800 D. H. Lehmer and Robert Canepa The snail lemma /entry/62 The classical snake lemma produces a six terms exact sequence starting from a commutative square with one of the edge being a regular epimorphism. We establish a new diagram lemma, that we call snail lemma, removing such a condition. We also show that the snail lemma subsumes the snake lemma and we give an interpretation of the snail lemma in terms of strong homotopy kernels. Our results hold in any pointed regular protomodular category. 62 Wed, 31 Dec 1969 16:33:36 -0800 Enrico M. Vitale Challenging mathematical problems with elementary solutions /entry/63 63 Wed, 31 Dec 1969 16:33:36 -0800 A.M. Yaglom and I.M. Yaglom On Pellegrino's 20-Caps in $S_{4,3}$ /entry/64 Although Pellegrino demonstrated that every 20-cap in $S_{4,3}$ is one of two geometric types, but it is by no means clear how many inequivalent 20-caps are there in each type. This chapter demonstrates that there are in all exactly nine inequivalent 20-caps in $S_{4,3}$. It also shows that just two of these occur as the intersection of a 56-cap in $S_{5,3}$ with a hyperplane. Because any 10-cap in $S_{3,3}$ is an elliptic quadric and is unique up to equivalence, it follows that any choice of E and V is equivalent to any other. However, for a given choice of E and V, there are 310 different r-caps. The seemingly difficult task of finding how many of these are inequivalent is made relatively simple by using the triple transitivity of the group Aut E on the points of E, together with the uniqueness of the ternary Golay code. The chapter identifies those 20-caps that occur as the intersection of a 56-cap in $S_{5,3}$ with a hyperplane and shows that caps of both these types do occur as sections of a 56-cap in $S_{5,3}$. 64 Wed, 31 Dec 1969 16:33:36 -0800 R. Hill Counting groups: gnus, moas and other exotica /entry/65 The number of groups of a given order is a fascinating function. We report on its known values, discuss some of its properties, and study some related functions. 65 Wed, 31 Dec 1969 16:33:36 -0800 John H. Conway and Heiko Dietrich and E.A. O’Brien On the Cookie Monster Problem /entry/66 The Cookie Monster Problem supposes that the Cookie Monster wants to empty a set of jars filled with various numbers of cookies. On each of his moves, he may choose any subset of jars and take the same number of cookies from each of those jars. The Cookie Monster number of a set is the minimum number of moves the Cookie Monster must use to empty all of the jars. This number depends on the initial distribution of cookies in the jars. We discuss bounds of the Cookie Monster number and explicitly find the Cookie Monster number for jars containing cookies in the Fibonacci, Tribonacci, n-nacci, and Super-n-nacci sequences. We also construct sequences of k jars such that their Cookie Monster numbers are asymptotically rk, where r is any real number between 0 and 1 inclusive. 66 Wed, 31 Dec 1969 16:33:36 -0800 Leigh Marie Braswell and Tanya Khovanova Two notes on notation /entry/67 The author advocates two specific mathematical notations from his popular course and joint textbook, "Concrete Mathematics". The first of these, extending an idea of Iverson, is the notation "[P]" for the function which is 1 when the Boolean condition P is true and 0 otherwise. This notation can encourage and clarify the use of characteristic functions and Kronecker deltas in sums and integrals. The second notation puts Stirling numbers on the same footing as binomial coefficients. Since binomial coefficients are written on two lines in parentheses and read "n choose k", Stirling numbers of the first kind should be written on two lines in brackets and read "n cycle k", while Stirling numbers of the second kind should be written in braces and read "n subset k". (I might say "n partition k".) The written form was first suggested by Imanuel Marx. The virtues of this notation are that Stirling partition numbers frequently appear in combinatorics, and that it more clearly presents functional relations similar to those satisfied by binomial coefficients. 67 Wed, 31 Dec 1969 16:33:36 -0800 Donald E. Knuth Dismal Arithmetic /entry/68 Dismal arithmetic is just like the arithmetic you learned in school, only simpler: there are no carries, when you add digits you just take the largest, and when you multiply digits you take the smallest. This paper studies basic number theory in this world, including analogues of the primes, number of divisors, sum of divisors, and the partition function. 68 Wed, 31 Dec 1969 16:33:36 -0800 David Applegate and Marc LeBrun and N. J. A. Sloane Fibonacci Jigsaw Puzzle /entry/69 69 Wed, 31 Dec 1969 16:33:36 -0800 Akio Hizume Prime numbers in certain arithmetic progressions /entry/70 We discuss to what extent Euclid's elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet's theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod $k$) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields. 70 Wed, 31 Dec 1969 16:33:36 -0800 Ram Murty and Nithum Thain Division by zero /entry/71 As a consequence of the MRDP theorem, the set of Diophantine equations provably unsolvable in any sufficiently strong theory of arithmetic is algorithmically undecidable. In contrast, we show the decidability of Diophantine equations provably unsolvable in Robinson's arithmetic Q. The argument hinges on an analysis of a particular class of equations, hitherto unexplored in Diophantine literature. We also axiomatize the universal fragment of Q in the process. 71 Wed, 31 Dec 1969 16:33:36 -0800 Je\vrábek, Emil Transposable integers in arbitrary bases /entry/72 72 Wed, 31 Dec 1969 16:33:36 -0800 Anne L. Ludington A Dozen Hat Problems /entry/73 73 Wed, 31 Dec 1969 16:33:36 -0800 Ezra Brown and James Tanton De Bruijn's Combinatorics /entry/74 This is a translation of the handwritten classroom notes taken by Nienhuys of a course in combinatorics given by N.G. de Bruijn at Eindhoven University of Technology, during the 1970s and 1980s. 74 Wed, 31 Dec 1969 16:33:36 -0800 Hung, J.W.Nienhuys (Ling-Ju and Eds.), Ton Kloks On gardeners, dukes and mathematical instruments /entry/75 Postprint (author's final draft) 75 Wed, 31 Dec 1969 16:33:36 -0800 Blanco Abellán, Mónica Comparative kinetics of the snowball respect to other dynamical objects /entry/76 We examine the kinetics of a snowball that is gaining mass while is rolling downhill. This dynamical system combines rotational effects with effects involving the variation of mass. In order to understand the consequences of both effects we compare its behavior with the one of some objects in which such effects are absent. Environmental conditions are also included. We conclude that the comparative velocity of the snowball is very sensitive to the hill profile and the retardation factors. We emphasize that the increase of mass (inertia), could surprisingly diminish the retardation effect due to the drag force. Additionally, when an exponential trajectory is assumed, the maximum velocity of the snowball can be reached at an intermediate step of the trip. 76 Wed, 31 Dec 1969 16:33:36 -0800 Diaz, Rodolfo A. and Gonzalez, Diego L. and Marin, Francisco and Martinez, R. Area and Hausdorff Dimension of Julia Sets of Entire Functions /entry/77 We show the Julia set of $\lambda \sin(z)$ has positive area and the action of $\lambda \sin(z)$ on its Julia set is not ergodic; the Julia set of $\lambda \exp(z)$ has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero. 77 Wed, 31 Dec 1969 16:33:35 -0800 Curt McMullen Approaches to the Enumerative Theory of Meanders /entry/78 78 Wed, 31 Dec 1969 16:33:35 -0800 Michael La Croix The Theory of Heaps and the Cartier-Foata Monoid /entry/79 We present Viennot’s theory of heaps of pieces, show that heaps are equivalent to elements in the partially commutative monoid of Cartier and Foata, and illustrate the main results of the theory by reproducing its application to the enumeration of parallelogram polyominoes due to Bousquet–Mélou and Viennot. 79 Wed, 31 Dec 1969 16:33:35 -0800 C. Krattenthaler Planar graph is on fire /entry/80 Let $G$ be any connected graph on $n$ vertices, $n \ge 2.$ Let $k$ be any positive integer. Suppose that a fire breaks out on some vertex of $G.$ Then in each turn $k$ firefighters can protect vertices of $G$ --- each can protect one vertex not yet on fire; Next a fire spreads to all unprotected neighbours. The $\$emph{$k$-surviving} rate of G, denoted by $\rho_k(G),$ is the expected fraction of vertices that can be saved from the fire by $k$ firefighters, provided that the starting vertex is chosen uniformly at random. In this paper, it is shown that for any planar graph $G$ we have $\rho_3(G) \ge \frac{2}{21}.$ Moreover, 3 firefighters are needed for the first step only; after that it is enough to have 2 firefighters per each round. This result significantly improves known solutions to a problem of Cai and Wang (there was no positive bound known for surviving rate of general planar graph with only 3 firefighters). The proof is done using the separator theorem for planar graphs. 80 Wed, 31 Dec 1969 16:33:35 -0800 Gordinowicz, Przemysław On the Number of Times an Integer Occurs as a Binomial Coefficient /entry/81 81 Wed, 31 Dec 1969 16:33:35 -0800 H. L. Abbott and P. Erdős and D. Hanson What to do when the trisector comes /entry/82 82 Wed, 31 Dec 1969 16:33:35 -0800 Dudley, Underwood The effective content of Reverse Nonstandard Mathematics and the nonstandard content of effective Reverse Mathematics /entry/83 The aim of this paper is to highlight a hitherto unknown computational aspect of Nonstandard Analysis pertaining to Reverse Mathematics (RM). In particular, we shall establish RM-equivalences between theorems from Nonstandard Analysis in a fragment of Nelson's internal set theory. We then extract primitive recursive terms from Goedel's system T (not involving Nonstandard Analysis) from the proofs of the aforementioned nonstandard equivalences. The resulting terms turn out to be witnesses for effective1 equivalences in Kohlenbach's higher-order RM. In other words, from an RM-equivalence in Nonstandard Analysis, we can extract the associated effective higher-order RM-equivalence which does not involve Nonstandard Analysis anymore. Finally, we show that certain effective equivalences in turn give rise to the original nonstandard theorems from which they were derived. 83 Wed, 31 Dec 1969 16:33:35 -0800 Sanders, Sam When is .999... less than 1? /entry/84 We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone's "semicolon" notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol .999 . . . be inculcated in a pre-limit teaching environment? The problem of the unital evaluation is hereby examined from the pre-R, pre-lim viewpoint of the student. 84 Wed, 31 Dec 1969 16:33:35 -0800 Katz, Karin Usadi and Katz, Mikhail G. Haruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite /entry/85 While directed site-animals have been solved on several lattices, directed bond-animals remain unsolved on any nontrivial lattice. In this paper we demonstrate that the anisotropic generating function of directed bond-animals on the square lattice is fundamentally different from that of directed site-animals in that it is not differentiably finite. We also extend this result to directed bond-animals on hypercubic lattices. This indicates that directed bond-animals are unlikely to be solved by similar methods to those used in the solution of directed site-animals. It also implies that a solution cannot be conjectured using computer packages such as Gfun [A Maple package developed by B. Salvy, P. Zimmermann, E. Murray at INRIA, France, available from http://algo.inria.fr/libraries/ at time of submission; B. Salvy, P. Zimmermann, Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (2) (1994) 163–177] or differential approximants [A.J. Guttmann, Asymptotic analysis of coefficients, in: C. Domb, J. Lebowitz (Eds.), Phase Transit. Crit. Phenom., vol. 13, Academic Press, London, 1989, pp. 1–234, programs available from http://www.ms.unimelb.edu.au/~tonyg]. 85 Wed, 31 Dec 1969 16:33:35 -0800 Rechnitzer, Andrew Haruspicy and anisotropic generating functions /entry/86 Guttmann and Enting [Phys. Rev. Lett. 76 (1996) 344–347] proposed the examination of anisotropic generating functions as a test of the solvability of models of bond animals. In this article we describe a technique for examining some properties of anisotropic generating functions. For a wide range of solved and unsolved families of bond animals, we show that the coefficients of yn is rational, the degree of its numerator is at most that of its denominator, and the denominator is a product of cyclotomic polynomials. Further, we are able to find a multiplicative upper bound for these denominators which, by comparison with numerical studies [Jensen, personal communication; Jensen and Guttmann, personal communication], appears to be very tight. These facts can be used to greatly reduce the amount of computation required in generating series expansions. They also have strong and negative implications for the solvability of these problems. 86 Wed, 31 Dec 1969 16:33:35 -0800 Rechnitzer, Andrew Spiralling self-avoiding walks: an exact solution /entry/87 87 Wed, 31 Dec 1969 16:33:35 -0800 Blote, H W J and Hilhorst, H J How to Beat Your Wythoff Games' Opponent on Three Fronts /entry/88 88 Wed, 31 Dec 1969 16:33:35 -0800 Aviezri S. Fraenkel Dice - Numericana /entry/89 89 Wed, 31 Dec 1969 16:33:35 -0800 Gérard P. Michon Proposal to Encode the Ganda Currency Mark for Bengali in ISO/IEC 10646 /entry/90 90 Wed, 31 Dec 1969 16:33:35 -0800 Anshuman Pandey Another Proof of Segre's Theorem about Ovals /entry/91 In 1955 B. Segre showed that any oval in a projective plane over a finite field of odd order is a conic. His proof constructs a conic which matches the oval in some points, and then shows that it actually coincides with the oval. Here we give another proof. We describe the oval by a possibly high degree polynomial, and then show that the degree is actually 2. 91 Wed, 31 Dec 1969 16:33:35 -0800 Müller, Peter Fair Dice /entry/92 92 Wed, 31 Dec 1969 16:33:35 -0800 Diaconis, Persi and Keller, Joseph B On the Existence of Generalized Parking Spaces for Complex Reflection Groups /entry/93 Let $W$ be an irreducible finite complex reflection group acting on a complex vector space $V$. For a positive integer $k$, we consider a class function $\varphi_k$ given by $\varphi_k(w) = k^{\dim V^w}$ for $w \in W$, where $V^w$ is the fixed-point subspace of $w$. If $W$ is the symmetric group of $n$ letters and $k=n+1$, then $\varphi_{n+1}$ is the permutation character on (classical) parking functions. In this paper, we give a complete answer to the question when $\varphi_k$ (resp. its $q$-analogue) is the character of a representation (resp. the graded character of a graded representation) of $W$. As a key to the proof in the symmetric group case, we find the greatest common divisors of specialized Schur functions. And we propose a unimodality conjecture of the coefficients of certain quotients of principally specialized Schur functions. 93 Wed, 31 Dec 1969 16:33:35 -0800 Ito, Yosuke and Okada, Soichi Mind the Croc! Rationality Gaps vis-à-vis the Crocodile Paradox /entry/94 This article discusses rationality gaps triggered by self-referential/cyclic choice, the latter being understood as choosing according to a norm that refers to the choosing itself. The Crocodile Paradox is reformulated and analyzed as a game—named CP—whose Nash equilibrium is shown to trigger a cyclic choice and to invite a rationality gap. It is shown that choosing the Nash equilibrium of CP conforms to the principles Wolfgang Spohn and Haim Gaifman introduced to, allegedly, guarantee acyclicity but, in fact, does not prevent self-referential/cyclic choice and rationality gaps. It is shown that CP is a counter-example to Gaifman's solution of the rationality gaps problem. 94 Wed, 31 Dec 1969 16:33:35 -0800 Gerogiorgakis, Stamatios Denser Egyptian Fractions /entry/95 An Egyptian fraction is a sum of distinct unit fractions (reciprocals of positive integers). We show that every rational number has Egyptian fraction representations where the number of terms is of the same order of magnitude as the largest denominator, improving a result from an earlier paper to best-possible form. We also settle, in best-possible form, the related problem of how small M_t(r) can be such that there is an Egyptian fraction representation of r with exactly t terms, the denominators of which are all at most M_t(r). We also consider the following problems posed by Erdős and Graham: the set of integers that cannot be the largest denominator of an Egyptian fraction representation of 1 is infinite - what is its order of growth? How about those integers that cannot be the second-largest (third-largest, etc.) denominator of such a representation? In the latter case, we show that only finitely many integers cannot be the second-largest (third-largest, etc.) denominator of such a representation; while in the former case, we show that the set of integers that cannot be the largest denominator of such a representation has density zero, and establish its order of growth. Both results extend to representations of any positive rational number. 95 Wed, 31 Dec 1969 16:33:35 -0800 Martin, Greg Reversible quantum cellular automata /entry/96 We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules also generate the global time step for automata with periodic boundary conditions. Our main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme. This implies that, in contrast to the classical case, the inverse of a nearest neighbor quantum cellular automaton is again a nearest neighbor automaton. We present several construction methods for quantum cellular automata, based on unitaries commuting with their translates, on the quantization of (arbitrary) reversible classical cellular automata, on quantum circuits, and on Clifford transformations with respect to a description of the single cells by finite Weyl systems. Moreover, we indicate how quantum random walks can be considered as special cases of cellular automata, namely by restricting a quantum lattice gas automaton with local particle number conservation to the single particle sector. 96 Wed, 31 Dec 1969 16:33:35 -0800 Schumacher, B. and Werner, R. F. Representations of Palindromic, Prime and Number Patterns /entry/97 97 Wed, 31 Dec 1969 16:33:35 -0800 Inder J. Taneja Have you been using the wrong estimator? These guys bound average fidelity using this one weird trick von Neumann didn't want you to know /entry/98 We give bounds on the average fidelity achievable by any quantum state estimator, which is arguably the most prominently used figure of merit in quantum state tomography. Moreover, these bounds can be computed online---that is, while the experiment is running. We show numerically that these bounds are quite tight for relevant distributions of density matrices. We also show that the Bayesian mean estimator is ideal in the sense of performing close to the bound without requiring optimization. Our results hold for all finite dimensional quantum systems. 98 Wed, 31 Dec 1969 16:33:35 -0800 Ferrie, Christopher and Kueng, Richard The Lost Calculus (1637-1670): Tangency and Optimization without Limits /entry/99 An examination of the evolution of the lost calculus from its beginnings in the work of Descartes and its subsequent development by Hudde, and the possibility that nearly every problem of calculus could have been solved using algorithms entirely free from the limit concept. 99 Wed, 31 Dec 1969 16:33:35 -0800 Jeff Suzuki The accuracy of Buffon's needle: a rule of thumb used by ants to estimate area /entry/100 Colonies of the ant Leptothorax albipennis naturally inhabit flat rock crevices. Scouts can determine, before initiating an emigration, if a nest has sufficient area to house their colony. They do so with a rule of thumb: the Buffon's needle algorithm. Based on a derivation from the classical statistical geometry of Comte George de Buffon in the 18th century, it can be shown that it is possible to estimate the area of a plane from the frequency of intersections between two sets of randomly scattered lines of known lengths. Our earlier work has shown that individual ants use this Buffon's needle algorithm by laying individual-specific trail pheromones on a first visit to a potential nest site and by assessing the frequency at which they intersect that path on a second visit. Nest area would be inversely proportional to the intersection frequency. The simplest procedure would be for individual ants to keep their first-visit path-length constant regardless of the size of the nest they are visiting. Here we show, for the first time, that this is the case. We also determine the potential quality of information that individual ants might have at their disposal from their own path-laying and path-crossing activities. Hence, we can determine the potential accuracy of nest area estimation by individual ants. Our findings suggest that ants using the Buffon's needle rule of thumb might obtain remarkably accurate assessments of nest area. 100 Wed, 31 Dec 1969 16:33:35 -0800 Mugford, S. T. Finding long chains in kidney exchange using the traveling salesman problem /entry/101 SignificanceThere are currently more than 100,000 patients on the waiting list in the United States for a kidney transplant from a deceased donor. To address this shortage, kidney exchange programs allow patients with living incompatible donors to exchange donors through cycles and chains initiated by altruistic nondirected donors. To determine which exchanges will take place, kidney exchange programs use algorithms for maximizing the number of transplants under constraints about the size of feasible exchanges. This problem is NP-hard, and algorithms previously used were unable to optimize when chains could be long. We developed two algorithms that use integer programming to solve this problem, one of which is inspired by the traveling salesman, that together can find optimal solutions in practice. As of May 2014 there were more than 100,000 patients on the waiting list for a kidney transplant from a deceased donor. Although the preferred treatment is a kidney transplant, every year there are fewer donors than new patients, so the wait for a transplant continues to grow. To address this shortage, kidney paired donation (KPD) programs allow patients with living but biologically incompatible donors to exchange donors through cycles or chains initiated by altruistic (nondirected) donors, thereby increasing the supply of kidneys in the system. In many KPD programs a centralized algorithm determines which exchanges will take place to maximize the total number of transplants performed. This optimization problem has proven challenging both in theory, because it is NP-hard, and in practice, because the algorithms previously used were unable to optimally search over all long chains. We give two new algorithms that use integer programming to optimally solve this problem, one of which is inspired by the techniques used to solve the traveling salesman problem. These algorithms provide the tools needed to find optimal solutions in practice. 101 Wed, 31 Dec 1969 16:33:35 -0800 Anderson, Ross and Ashlagi, Itai and Gamarnik, David and Roth, Alvin E. Maximum Matching and a Polyhedron With 0,1-Vertices /entry/102 A matching in a graph $G$ is a subset of edges in $G$ such that no two meet the same node in $G$. The convex polyhedron $C$ is characterised, where the extreme points of $C$ correspond to the matchings in $G$. Where each edge of $G$ carries a real numerical weight, an efficient algorithm is described for finding a matching in $G$ with maximum weight-sum. 102 Wed, 31 Dec 1969 16:33:35 -0800 Jack Edmonds Magic squares of seventh powers /entry/103 103 Wed, 31 Dec 1969 16:33:35 -0800 Christian Boyer On Legendre's Prime Number Formula /entry/104 104 Wed, 31 Dec 1969 16:33:35 -0800 Janos Pintz A Brief Critique of Pure Hypercomputation /entry/105 105 Wed, 31 Dec 1969 16:33:35 -0800 Cotogno, Paolo Complexity and Algorithms for Graph and Hypergraph Sandwich Problems /entry/106 106 Wed, 31 Dec 1969 16:33:35 -0800 Golumbic, Martin Charles and Wassermann, Amir A combinatorial theorem in plane geometry /entry/107 107 Wed, 31 Dec 1969 16:33:35 -0800 Chvátal, V The dying rabbit problem revisited /entry/108 In this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in the paper Hoggatt, V. E., Jr.; Lind, D. A. "The dying rabbit problem". Fibonacci Quart. 7 1969 no. 5, 482--487) and we calculate explicitly their general term (extending the work in the paper Miles, E. P., Jr. Generalized Fibonacci numbers and associated matrices. Amer. Math. Monthly 67 1960 745--752). In passing, and as a technical requirement, we also study the behavior of the positive real roots of the characteristic polynomial of the considered sequences. 108 Wed, 31 Dec 1969 16:33:35 -0800 Oller, Antonio M. Rational approximations to $\pi$ and some other numbers /entry/109 109 Wed, 31 Dec 1969 16:33:35 -0800 Hata, Masayoshi and Mignotte, M and Chudnovsky, G V and Beukers, F Efficient Algorithms for Zeckendorf Arithmetic /entry/110 We study the problem of addition and subtraction using the Zeckendorf representation of integers. We show that both operations can be performed in linear time; in fact they can be performed by combinational logic networks with linear size and logarithmic depth. The implications of these results for multiplication, division and square-root extraction are also discussed. 110 Wed, 31 Dec 1969 16:33:35 -0800 Ahlbach, Connor and Usatine, Jeremy and Pippenger, Nicholas Exact Approximations of Omega Numbers /entry/111 111 Wed, 31 Dec 1969 16:33:35 -0800 Calude, C.S and Dinneen, Michael Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice /entry/112 We enumerate the symmetry classes of convex polyominoes on the hexagonal (honeycomb) lattice. Here convexity is to be understood as convexity along the three main column directions. We deduce the generating series of free (i.e. up to reflection and rotation) and of asymmetric convex hexagonal polyominoes, according to area and half-perimeter. We give explicit formulas or implicit functional equations for the generating series, which are convenient for computer algebra. Thus, computations can be carried out up to area 70. 112 Wed, 31 Dec 1969 16:33:35 -0800 Gouyou-Beauchamps, Dominique and Leroux, Pierre On dice and coins: Models of computation for random generation /entry/113 113 Wed, 31 Dec 1969 16:33:35 -0800 Feldman, D and Impagliazzo, R and Naor, M This is the (co)end, my only (co)friend /entry/114 The present note is a recollection of the most striking and useful applications of co/end calculus. We put a considerable effort in making arguments and constructions rather explicit: after having given a series of preliminary definitions, we characterize co/ends as particular co/limits; then we derive a number of results directly from this characterization. The last sections discuss the most interesting examples where co/end calculus serves as a powerful abstract way to do explicit computations in diverse fields like Algebra, Algebraic Topology and Category Theory. The appendices serve to sketch a number of results in theories heavily relying on co/end calculus; the reader who dares to arrive at this point, being completely introduced to the mysteries of co/end fu, can regard basically every statement as a guided exercise. 114 Wed, 31 Dec 1969 16:33:35 -0800 Loregian, Fosco The Eudoxus Real Numbers /entry/115 This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is inspired by the ancient Greek theory of proportion, due to Eudoxus. However, unlike the Dedekind construction, the construction proceeds directly from the integers to the real numbers bypassing the intermediate construction of the rational numbers. The construction of the additive group of the reals depends on rather simple algebraic properties of the integers. This part of the construction can be generalised in several natural ways, e.g., with an arbitrary abelian group playing the role of the integers. This raises the question: what does the construction construct? In an appendix we sketch some generalisations and answer this question in some simple cases. The treatment of the main construction is intended to be self-contained and assumes familiarity only with elementary algebra in the ring of integers and with the definitions of the abstract algebraic notions of group, ring and field. 115 Wed, 31 Dec 1969 16:33:35 -0800 Arthan, R. D. Bells, Motels and Permutation Groups /entry/116 This article is about the mathematics of ringing the changes. We describe the mathematics which arises from a real-world activity, that of ringing the changes on bells. We present Rankin's solution of one of the famous old problems in the subject. This article was written in 2003. 116 Wed, 31 Dec 1969 16:33:34 -0800 McGuire, Gary Music: a Mathematical Offering /entry/117 117 Wed, 31 Dec 1969 16:33:34 -0800 Dave Benson Irrationality From The Book /entry/118 We generalize Tennenbaum's geometric proof of the irrationality of sqrt(2) to sqrt(n) for n = 3, 5, 6 and 10. 118 Wed, 31 Dec 1969 16:33:34 -0800 Miller, Steven J. and Montague, David Division by three /entry/119 119 Wed, 31 Dec 1969 16:33:34 -0800 Doyle, Peter G. and Conway, John Horton How not to prove the Poincaré conjecture /entry/120 I have committed the sin of falsely proving Poincaré's Conjecture. But that was in another country; and besides, until now no one has known about it. Now, in hope of deterring others from making similar mistakes, I shall describe my mistaken proof. Who knows but that somehow a small change, a new interpretation, and this line of proof may be rectified! 120 Wed, 31 Dec 1969 16:33:34 -0800 Stallings, JR An Application of Elementary Group Theory to Central Solitaire /entry/121 121 Wed, 31 Dec 1969 16:33:34 -0800 Bialostocki, Arie Solving Triangular Peg Solitaire /entry/122 We consider the one-person game of peg solitaire on a triangular board of arbitrary size. The basic game begins from a full board with one peg missing and finishes with one peg at a specified board location. We develop necessary and sufficient conditions for this game to be solvable. For all solvable problems, we give an explicit solution algorithm. On the 15-hole board, we compare three simple solution strategies. We then consider the problem of finding solutions that minimize the number of moves (where a move is one or more consecutive jumps by the same peg), and find the shortest solution to the basic game on all triangular boards with up to 55 holes (10 holes on a side). 122 Wed, 31 Dec 1969 16:33:34 -0800 Bell, George I. The Super Patalan Numbers /entry/123 We introduce the super Patalan numbers, a generalization of the super Catalan numbers in the sense of Gessel, and prove a number of properties analagous to those of the super Catalan numbers. The super Patalan numbers generalize the super Catalan numbers similarly to how the Patalan numbers generalize the Catalan numbers. 123 Wed, 31 Dec 1969 16:33:34 -0800 Richardson, Thomas M. The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them? /entry/124 124 Wed, 31 Dec 1969 16:33:34 -0800 Durán, Antonio J and Pérez, Mario and Varona, Juan L Proofs without syntax /entry/125 125 Wed, 31 Dec 1969 16:33:34 -0800 Hughes, DJD Methods for studying coincidences /entry/126 This article illustrates basic statistical techniques for studying coincidences. These include data-gathering methods (informal anecdotes, case studies, observational studies, and experiments) and methods of analysis (exploratory and confirmatory data analysis, special analytic techniques, and probabilistic modeling, both general and special purpose). We develop a version of the birthday problem general enough to include dependence, inhomogeneity, and almost multiple matches. We review Fisher’s techniques for giving partial credit for close matches. We develop a model for studying coincidences involving newly learned words. Once we set aside coincidences having apparent causes, four principles account for large numbers of remaining coincidences: hidden cause; psychology, including memory and perception; multiplicity of endpoints, including the counting of “close” or nearly alike events as if they were identical; and the law of truly large numbers which says that when enormous numbers of events and people and their interactions cumulate over time, almost any outrageous event is bound to occur. These sources account for much of the force of synchronicity. 126 Wed, 31 Dec 1969 16:33:34 -0800 Diaconis, P and Mosteller, Frederick Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes /entry/127 127 Wed, 31 Dec 1969 16:33:34 -0800 Bailey, RA How often should you clean your room? /entry/128 We introduce and study a combinatorial optimization problem motivated by the question in the title. In the simple case where you use all objects in your room equally often, we investigate asymptotics of the optimal time to clean up in terms of the number of objects in your room. In particular, we prove a logarithmic upper bound, solve an approximate version of this problem, and conjecture a precise logarithmic asymptotic. 128 Wed, 31 Dec 1969 16:33:34 -0800 Martin, Kimball and Shankar, Krishnan Pondering an Artist's Perplexing Tribute to the Pythagorean Theorem /entry/129 129 Wed, 31 Dec 1969 16:33:34 -0800 Ivars Peterson A Fresh Look at Peg Solitaire /entry/130 130 Wed, 31 Dec 1969 16:33:34 -0800 George I. Bell The shape of a Mobius band /entry/131 131 Wed, 31 Dec 1969 16:33:34 -0800 Mahadevan, L and Keller, JB Foldings and Meanders /entry/132 We review the stamp folding problem, the number of ways to fold a strip of $n$ stamps, and the related problem of enumerating meander configurations. The study of equivalence classes of foldings and meanders under symmetries allows to characterize and enumerate folding and meander shapes. Symmetric foldings and meanders are described, and relations between folding and meandric sequences are given. Extended tables for these sequences are provided. 132 Wed, 31 Dec 1969 16:33:34 -0800 Legendre, Stéphane Mathematics and group theory in music /entry/133 The purpose of this paper is to show through particular examples how group theory is used in music. The examples are chosen from the theoretical work and from the compositions of Olivier Messiaen (1908-1992), one of the most influential twentieth century composers and pedagogues. Messiaen consciously used mathematical concepts derived from symmetry and groups, in his teaching and in his compositions. Before dwelling on this, I will give a quick overview of the relation between mathematics and music. This will put the discussion on symmetry and group theory in music in a broader context and it will provide the reader of this handbook some background and some motivation for the subject. The relation between mathematics and music, during more than two millennia, was lively, widespread, and extremely enriching for both domains. This paper will appear in the Handbook of Group actions, vol. II (ed. L. Ji, A. Papadopoulos and S.-T. Yau), Higher Eucation Press and International Press. 133 Wed, 31 Dec 1969 16:33:34 -0800 Papadopoulos, Athanase An arctic circle theorem for groves /entry/134 In earlier work, Jockusch, Propp, and Shor proved a theorem describing the limiting shape of the boundary between the uniformly tiled corners of a random tiling of an Aztec diamond and the more unpredictable `temperate zone' in the interior of the region. The so-called arctic circle theorem made precise a phenomenon observed in random tilings of large Aztec diamonds. Here we examine a related combinatorial model called groves. Created by Carroll and Speyer as combinatorial interpretations for Laurent polynomials given by the cube recurrence, groves have observable frozen regions which we describe precisely via asymptotic analysis of a generating function. Our approach also provides another way to prove the arctic circle theorem for Aztec diamonds. 134 Wed, 31 Dec 1969 16:33:34 -0800 Petersen, T. K. and Speyer, D. History-dependent random processes /entry/135 Ulam has defined a history-dependent random sequence by the recursion Xn+1=Xn+XU(n), where (U(n); n[&ge;]1) is a sequence of independent random variables with U(n) uniformly distributed on {1, ..., n} and X1=1. We introduce a new class of continuous-time history-dependent random processes regulated by Poisson processes. The simplest of these, a univariate process regulated by a homogeneous Poisson process, replicates in continuous time the essential properties of Ulam's sequence, and greatly facilitates its analysis. We consider several generalizations and extensions of this, including bivariate and multivariate coupled history-dependent processes, and cases when the dependence on the past is not uniform. The analysis of the discrete-time formulations of these models would be at the very least an extremely formidable project, but we determine the asymptotic growth rates of their means and higher moments with relative ease. 135 Wed, 31 Dec 1969 16:33:34 -0800 Clifford, P. and Stirzaker, D. LIM is not slim /entry/136 In this paper LIM, a recently proposed impartial combinatorial ruleset, is analyzed. A formula to describe the $G$-values of LIM positions is given, by way of analyzing an equivalent combinatorial ruleset LIM’, closely related to the classical nim. Also, an enumeration of $P$-positions of LIM with $n$ stones, and its relation to the Ulam-Warburton cellular automaton, is presented. 136 Wed, 31 Dec 1969 16:33:34 -0800 Fink, Alex and Fraenkel, Aviezri S. and Santos, Carlos The Number-Pad Game /entry/137 137 Wed, 31 Dec 1969 16:33:34 -0800 Alex Fink and Richard Guy Nim Fractals /entry/138 We enumerate P-positions in the game of Nim in two different ways. In one series of sequences we enumerate them by the maximum number of counters in a pile. In another series of sequences we enumerate them by the total number of counters. We show that the game of Nim can be viewed as a cellular automaton, where the total number of counters divided by 2 can be considered as a generation in which P-positions are born. We prove that the three-pile Nim sequence enumerated by the total number of counters is a famous toothpick sequence based on the Ulam-Warburton cellular automaton. We introduce 10 new sequences. 138 Wed, 31 Dec 1969 16:33:34 -0800 Khovanova, Tanya and Xiong, Joshua Generalizing Zeckendorf's Theorem to f-decompositions /entry/139 A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, where $F_1 = 1$, $F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$. For general recurrences $\{G_n\}$ with non-negative coefficients, there is a notion of a legal decomposition which again leads to a unique representation, and the number of summands in the representations of uniformly randomly chosen $m \in [G_n, G_{n+1})$ converges to a normal distribution as $n \to \infty$. We consider the converse question: given a notion of legal decomposition, is it possible to construct a sequence $\{a_n\}$ such that every positive integer can be decomposed as a sum of terms from the sequence? We encode a notion of legal decomposition as a function $f:\N_0\to\N_0$ and say that if $a_n$ is in an "$f$-decomposition", then the decomposition cannot contain the $f(n)$ terms immediately before $a_n$ in the sequence; special choices of $f$ yield many well known decompositions (including base-$b$, Zeckendorf and factorial). We prove that for any $f:\N_0\to\N_0$, there exists a sequence $\{a_n\}_{n=0}^\infty$ such that every positive integer has a unique $f$-decomposition using $\{a_n\}$. Further, if $f$ is periodic, then the unique increasing sequence $\{a_n\}$ that corresponds to $f$ satisfies a linear recurrence relation. Previous research only handled recurrence relations with no negative coefficients. We find a function $f$ that yields a sequence that cannot be described by such a recurrence relation. Finally, for a class of functions $f$, we prove that the number of summands in the $f$-decomposition of integers between two consecutive terms of the sequence converges to a normal distribution. 139 Wed, 31 Dec 1969 16:33:34 -0800 Demontigny, Philippe and Do, Thao and Kulkarni, Archit and Miller, Steven J. and Moon, David and Varma, Umang Useful inequalities cheat sheet /entry/140 This is a collection of some of the most important mathematical inequalities. I tried to include non-trivial inequalities that can be useful in solving problems or proving theorems. I omitted many details, in some cases even necessary conditions (hopefully only when they were obvious). If you are not sure whether an inequality can be applied in some context, try to find a more detailed source for the exact definition. For lack of space I omitted proofs and discussions on when equality holds. 140 Wed, 31 Dec 1969 16:33:34 -0800 László Kozma A Mathematical Coloring Book /entry/141 141 Wed, 31 Dec 1969 16:33:34 -0800 Hampton, Marshall On the diagram of 132-avoiding permutations /entry/142 142 Wed, 31 Dec 1969 16:33:34 -0800 Reifegerste, Astrid A number system with an irrational base /entry/143 143 Wed, 31 Dec 1969 16:33:34 -0800 George Bergman Eponymy in Mathematical Nomenclature: What's in a Name, and What Should Be? /entry/144 144 Wed, 31 Dec 1969 16:33:34 -0800 Henwood, Mervyn R. and Rival, Ivan Table for Fundamentals of Series : Part I : Basic Properties of Series and Products /entry/145 145 Wed, 31 Dec 1969 16:33:34 -0800 Gould, Henry W. More ties than we thought /entry/146 We extend the existing enumeration of neck tie knots to include tie knots with a textured front, tied with the narrow end of a tie. These tie knots have gained popularity in recent years, based on reconstructions of a costume detail from The Matrix Reloaded, and are explicitly ruled out in the enumeration by Fink and Mao (2000). We show that the relaxed tie knot description language that comprehensively describes these extended tie knot classes is either context sensitive or context free. It has a sub-language that covers all the knots that inspired the work, and that is regular. From this regular sub-language we enumerate 177 147 distinct tie knots that seem tieable with a normal necktie. These are found through an enumeration of 2 046 winding patterns that can be varied by tucking the tie under itself at various points along the winding. 146 Wed, 31 Dec 1969 16:33:34 -0800 Hirsch, Dan and Patterson, Meredith L and Sandberg, Anders and Vejdemo-Johansson, Mikael Rithmomachia /entry/147 This complex chess-like game appeared in the western world around the year 1000. The game knew a great burst of popularity in the 15th century, because of some rules changes. When chess also saw its rules change (particularly when the Queen started to move in its modern fashion instead of its previous King-like motion), Rithmomachia started fading rapidly, at the close of the 16th century. The rules given here are those established in 1556 by Claude de Boissière, a Frenchman. 147 Wed, 31 Dec 1969 16:33:34 -0800 Daniel U. Thibault and Michel Boutin Mathematical Games /entry/148 148 Wed, 31 Dec 1969 16:33:34 -0800 Silva, Jorge Nuno Linear recurrences through tilings and Markov chains /entry/149 149 Wed, 31 Dec 1969 16:33:34 -0800 Benjamin, AT and Hanusa, CRH and Su, FE The Stick Problem /entry/150 150 Wed, 31 Dec 1969 16:33:34 -0800 Augustine Bertagnolli Circular orbits on a warped spandex fabric /entry/151 We present a theoretical and experimental analysis of circular-like orbits made by a marble rolling on a warped spandex fabric. We show that the mass of the fabric interior to the orbital path influences the motion of the marble in a nontrivial way, and can even dominate the orbital characteristics. We also compare a Kepler-like expression for such orbits to similar expressions for orbits about a spherically-symmetric massive object in the presence of a constant vacuum energy, as described by general relativity. 151 Wed, 31 Dec 1969 16:33:34 -0800 Middleton, Chad A. and Langston, Michael The topology of competitively constructed graphs /entry/152 We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for this game: for the case $k=3$ a player can ensure the resulting graph is planar, while for the case $k=4$, a player can force the appearance of arbitrarily large clique minors. 152 Wed, 31 Dec 1969 16:33:33 -0800 Frieze, Alan and Pegden, Wesley The mathematics of Septoku /entry/153 Septoku is a Sudoku variant invented by Bruce Oberg, played on a hexagonal grid of 37 cells. We show that up to rotations, reflections, and symbol permutations, there are only six valid Septoku boards. In order to have a unique solution, we show that the minimum number of given values is six. We generalize the puzzle to other board shapes, and devise a puzzle on a star-shaped board with 73 cells with six givens which has a unique solution. We show how this puzzle relates to the unsolved Hadwiger-Nelson problem in combinatorial geometry. 153 Wed, 31 Dec 1969 16:33:33 -0800 Bell, George I. Fair but irregular polyhedral dice /entry/154 154 Wed, 31 Dec 1969 16:33:33 -0800 Joseph O'Rourke Solving Differential Equations by Symmetry Groups /entry/155 155 Wed, 31 Dec 1969 16:33:33 -0800 John Starret A knowledge-based approach of connect-four /entry/156 A Shannon C-type strategy program, VICTOR, is written for Connect-Four, based on nine strategic rules. Each of these rules is proven to be correct, implying that conclusions made by VICTOR are correct. Using VICTOR, strategic rules where found which can be used by Black to at least draw the game, on each 7 (2n) board, provided that White does not start at the middle column, as well as on any 6 (2n) board. In combination with conspiracy-number search, search tables and depth-first search, VICTOR was able to show that White can win on the standard 7 6 board. Using a database of approximately half a million positions, VICTOR can play real time against opponents on the 7 6 board, always winning with White. 156 Wed, 31 Dec 1969 16:33:33 -0800 Allis, Victor WHAT IS Lehmer's number? /entry/157 157 Wed, 31 Dec 1969 16:33:33 -0800 Eriko Hironaka Analyse algébrique d'un scrutin /entry/158 158 Wed, 31 Dec 1969 16:33:33 -0800 Guilbaud, GT and Rosenstiehl, P Lone Axes in Outer Space /entry/159 Handel and Mosher define the axis bundle for a fully irreducible outer automorphism in "Axes in Outer Space." In this paper we give a necessary and sufficient condition for the axis bundle to consist of a unique periodic fold line. As a consequence, we give a setting, and means for identifying in this setting, when two elements of an outer automorphism group $Out(F_r)$ have conjugate powers. 159 Wed, 31 Dec 1969 16:33:33 -0800 Mosher, Lee and Pfaff, Catherine The Maximum Throughput Rate for Each Hole on a Golf Course /entry/160 160 Wed, 31 Dec 1969 16:33:33 -0800 Whitt, Ward Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012) /entry/161 In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$ 1+\frac 12 +\frac 13+...+\frac{1}{p-1} $$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$ 1+\frac{1}{2^2} +\frac{1}{3^2}+...+\frac{1}{(p-1)^2} $$ written in reduced form is divisible by $p$. The first of the above congruences, the so called {\it Wolstenholme's theorem}, is a fundamental congruence in combinatorial number theory. In this article, consisting of 11 sections, we provide a historical survey of Wolstenholme's type congruences and related problems. Namely, we present and compare several generalizations and extensions of Wolstenholme's theorem obtained in the last hundred and fifty years. In particular, we present more than 70 variations and generalizations of this theorem including congruences for Wolstenholme primes. These congruences are discussed here by 33 remarks. The Bibliography of this article contains 106 references consisting of 13 textbooks and monographs, 89 papers, 3 problems and Sloane's On-Line Enc. of Integer Sequences. In this article, some results of these references are cited as generalizations of certain Wolstenholme's type congruences, but without the expositions of related congruences. The total number of citations given here is 189. 161 Wed, 31 Dec 1969 16:33:33 -0800 Mestrovic, Romeo The Math Encyclopedia of Smarandache Type Notions /entry/162 About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name) and literature, it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache's mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. We structured this book using numbered Definitions, Theorems, Conjectures, Notes and Comments, in order to facilitate an easier reading but also to facilitate references to a specific paragraph. We divided the Bibliography in two parts, Writings by Florentin Smarandache (indexed by the name of books and articles) and Writings on Smarandache notions (indexed by the name of authors). We treated, in this book, about 130 Smarandache type sequences, about 50 Smarandache type functions and many solved or open problems of number theory. We also have, at the end of this book, a proposal for a new Smarandache type notion, id est the concept of “a set of Smarandache-Coman divisors of order k of a composite positive integer n with m prime factors”, notion that seems to have promising applications, at a first glance at least in the study of absolute and relative Fermat pseudoprimes, Carmichael numbers and Poulet numbers. This encyclopedia is both for researchers that will have on hand a tool that will help them “navigate” in the universe of Smarandache type notions and for young math enthusiasts: many of them will be attached by this wonderful branch of mathematics, number theory, reading the works of Florentin Smarandache. 162 Wed, 31 Dec 1969 16:33:33 -0800 Coman, Marius 2178 And All That /entry/163 For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089 and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to study the problem of finding all (g,k)-reverse multiples. By using modified versions of her trees, which we call Young graphs, we determine the possible values of k for bases g = 2 through 100, and then show how to apply the transfer-matrix method to enumerate the (g,k)-reverse multiples with a given number of base-g digits. These Young graphs are interesting finite directed graphs, whose structure is not at all well understood. 163 Wed, 31 Dec 1969 16:33:33 -0800 Sloane, NJA Fibonacci numbers and Leonardo numbers /entry/164 164 Wed, 31 Dec 1969 16:33:33 -0800 Dijkstra, E.W. Pancake Flipping is Hard /entry/165 Pancake Flipping is the problem of sorting a stack of pancakes of different sizes (that is, a permutation), when the only allowed operation is to insert a spatula anywhere in the stack and to flip the pancakes above it (that is, to perform a prefix reversal). In the burnt variant, one side of each pancake is marked as burnt, and it is required to finish with all pancakes having the burnt side down. Computing the optimal scenario for any stack of pancakes and determining the worst-case stack for any stack size have been challenges over more than three decades. Beyond being an intriguing combinatorial problem in itself, it also yields applications, e.g. in parallel computing and computational biology. In this paper, we show that the Pancake Flipping problem, in its original (unburnt) variant, is NP-hard, thus answering the long-standing question of its computational complexity. 165 Wed, 31 Dec 1969 16:33:33 -0800 Bulteau, Laurent and Fertin, Guillaume and Rusu, Irena Giuga Numbers and the arithmetic derivative /entry/166 We characterize Giuga Numbers as solutions to the equation $n'=an+1$, with $a \in \mathbb{N}$ and $n'$ being the arithmetic derivative. Although this fact does not refute Lava's conjecture, it brings doubts about its veracity. 166 Wed, 31 Dec 1969 16:33:33 -0800 Grau, José María and Oller-Marcén, Antonio M. Swiss cheeses, rational approximation and universal plane curves /entry/167 167 Wed, 31 Dec 1969 16:33:33 -0800 Feinstein, JF and Heath, MJ Playing pool with $\pi$ (the number $\pi$ from a billiard point of view) /entry/168 168 Wed, 31 Dec 1969 16:33:33 -0800 Galperin, G Random Structures from Lego Bricks and Analog Monte Carlo Procedures /entry/169 169 Wed, 31 Dec 1969 16:33:33 -0800 Althöfer, I Is POPL Mathematics or Science? /entry/170 170 Wed, 31 Dec 1969 16:33:33 -0800 Andrew W. Appel Proofs by Descent /entry/171 171 Wed, 31 Dec 1969 16:33:33 -0800 CONRAD, K Practical numbers /entry/172 172 Wed, 31 Dec 1969 16:33:33 -0800 Srinivasan, A.K. How to differentiate a number /entry/173 173 Wed, 31 Dec 1969 16:33:33 -0800 Ufnarovski, Victor and Åhlander, B The Ubiquitous Thue-Morse Sequence /entry/174 174 Wed, 31 Dec 1969 16:33:33 -0800 Jeffrey Shallit Sloane's Gap: Do Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS? /entry/175 The Online Encyclopedia of Integer Sequences (OEIS) is made up of thousands of numerical sequences considered particularly interesting by some mathematicians. The graphic representation of the frequency with which a number n as a function of n appears in that database shows that the underlying function decreases fast, and that the points are distributed in a cloud, seemingly split into two by a clear zone that will be referred to here as "Sloane's Gap". The decrease and general form are explained by mathematics, but an explanation of the gap requires further considerations. 175 Wed, 31 Dec 1969 16:33:33 -0800 Gauvrit, Nicolas and Delahaye, Jean-Paul and Zenil, Hector Cookie Monster Devours Naccis /entry/176 In 2002, Cookie Monster appeared in The Inquisitive Problem Solver. The hungry monster wants to empty a set of jars filled with various numbers of cookies. On each of his moves, he may choose any subset of jars and take the same number of cookies from each of those jars. The Cookie Monster number is the minimum number of moves Cookie Monster must use to empty all of the jars. This number depends on the initial distribution of cookies in the jars. We discuss bounds of the Cookie Monster number and explicitly find the Cookie Monster number for Fibonacci, Tribonacci and other nacci sequences. 176 Wed, 31 Dec 1969 16:33:33 -0800 Braswell, Leigh Marie and Khovanova, Tanya The Strange and Surprising Saga of the Somos Sequences /entry/177 177 Wed, 31 Dec 1969 16:33:33 -0800 Gale, David An Infinite Set of Heron Triangles with Two Rational Medians /entry/178 178 Wed, 31 Dec 1969 16:33:33 -0800 Buchholz, Ralph H and Rathbun, Randall L The Laurent phenomenon /entry/179 A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of applications. In particular, we settle in the affirmative a conjecture of D.Gale and R.Robinson on integrality of generalized Somos sequences, and prove the Laurent property for several multidimensional recurrences, confirming conjectures by J.Propp, N.Elkies, and M.Kleber. 179 Wed, 31 Dec 1969 16:33:33 -0800 Fomin, Sergey and Zelevinsky, Andrei Perfect Matchings and the Octahedron Recurrence /entry/180 We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky. 180 Wed, 31 Dec 1969 16:33:33 -0800 Speyer, David E On n-Dimensional Polytope Schemes /entry/181 181 Wed, 31 Dec 1969 16:33:33 -0800 Fouhey, David F and Maturana, Daniel Only problems, not solutions! /entry/182 182 Wed, 31 Dec 1969 16:33:33 -0800 Smarandache, Florentin From Unicode to Typography, a Case Study the Greek Script /entry/183 183 Wed, 31 Dec 1969 16:33:33 -0800 Haralambous, Yannis Half of a coin: negative probabilities /entry/184 184 Wed, 31 Dec 1969 16:33:33 -0800 Székely, GJ Smooth neighbors /entry/185 We give a new algorithm that quickly finds smooth neighbors. 185 Wed, 31 Dec 1969 16:33:33 -0800 Conrey, Brian and Holmstrom, Mark and McLaughlin, Tara On a problem of Störmer /entry/186 186 Wed, 31 Dec 1969 16:33:33 -0800 Lehmer, DH Missing Data: Instrument-Level Heffalumps and Item-Level Woozles /entry/187 The purpose of this paper is to provide a brief overview of each of two missing data situations, and try to show the importance of considering which elusive creature a researcher might be hunting. We find that much of the previous literature does not consider the distinction between missing data at the item level or instrument level. Failure to make this distinction can partially muddle one’s treatment of missing data in important situations. 187 Wed, 31 Dec 1969 16:33:33 -0800 Philip L. Roth and Fred S. Switzer III Pascal's Pyramid Or Pascal's Tetrahedron /entry/188 A lattice of octahedra and tetrahedra (oct-tet lattice) is a useful paradigm for understanding the structure of Pascal's pyramid, the 3-D analog of Pascal's triangle. Notation for levels and coordinates of elements, a standard algorithm for generating the values of various elements, and a ratio method that is not dependent on the calculation of previous levels are discussed. Figures show a bell curve in 3 dimensions, the association of elements to primes and twin primes, and the values of elements mod(x) through patterns arranged in triangular plots. It is conjectured that the largest factor of any element is less than the level index. 188 Wed, 31 Dec 1969 16:33:33 -0800 Jim Nugent A Line of Sages /entry/189 189 Wed, 31 Dec 1969 16:33:33 -0800 Khovanova, Tanya Non-sexist solution of the ménage problem /entry/190 The ménage problem asks for the number of ways of seating \(n\) couples at a circular table, with men and women alternating, so that no one sits next to his or her partner. We present a straight-forward solution to this problem. What distinguishes our approach is that we do not seat the ladies first. 190 Wed, 31 Dec 1969 16:33:33 -0800 Kenneth P. Bogart The Ubiquitous Pi /entry/191 191 Wed, 31 Dec 1969 16:33:33 -0800 Castellanos, Dario Six Ways to Sum a Series /entry/192 A discussion of the sum of squares of the reciprocals of the positive integers with a review of several proofs. 192 Wed, 31 Dec 1969 16:33:33 -0800 Kalman, Dan Using Monoidal Categories in the Transformational Study of Musical Time-Spans and Rhythms /entry/193 Transformational musical theory has so far mainly focused on the study of groups acting on musical chords, one of the most famous example being the action of the dihedral group D24 on the set of major and minor chords. Comparatively less work has been devoted to the study of transformations of time-spans and rhythms. D. Lewin was the first to study group actions on time-spans by using a subgroup of the affine group in one dimension. In our previous work, the work of Lewin has been included in the more general framework of group extensions, and generalizations to time-spans on multiple timelines have been proposed. The goal of this paper is to show that such generalizations have a categorical background in free monodical categories generated by a group-as-category. In particular, symmetric monodical categories allow to deal with the possible interexchanges between timelines. We also show that more general time-spans can be considered, in which single time-spans are encapsulated in a "bracket" of time-spans, which allow for the description of complex rhythms. 193 Wed, 31 Dec 1969 16:33:33 -0800 Popoff, Alexandre Kindergarten Quantum Mechanics /entry/194 These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns `doing quantum mechanics using only pictures of lines, squares, triangles and diamonds'. This picture calculus can be seen as a very substantial extension of Dirac's notation, and has a purely algebraic counterpart in terms of so-called Strongly Compact Closed Categories (introduced by Abramsky and I in quant-ph/0402130 and [4]) which subsumes my Logic of Entanglement quant-ph/0402014. For a survey on the `what', the `why' and the `hows' I refer to a previous set of lecture notes quant-ph/0506132. In a last section we provide some pointers to the body of technical literature on the subject. 194 Wed, 31 Dec 1969 16:33:33 -0800 Coecke, Bob Cyclic twill-woven objects /entry/195 195 Wed, 31 Dec 1969 16:33:33 -0800 Akleman, Ergun and Chen, Jianer and Chen, YenLin and Xing, Qing and Gross, Jonathan L. Division of labor in child care: A game-theoretic approach /entry/196 196 Wed, 31 Dec 1969 16:33:33 -0800 Vierling-Claassen, a. A Do-It-Yourself Paper Digital Computer, 1959. /entry/197 This wonderful cut-away and paste-up template for a digital computer comes to us from the Communications of the Association for Computing Machinery, volume 2, issue 9 for September 1959. The PAPAC-00 is a “2-register, 1-bit, fixed-instruction binary digital computer” and was submitted to the journal by Rollin P. Mayer (of the MIT Lincoln Lab). 197 Wed, 31 Dec 1969 16:33:33 -0800 Ptak, John F. Familial sinistrals avoid exact numbers. /entry/198 We report data from an internet questionnaire of sixty number trivia. Participants were asked for the number of cups in their house, the number of cities they know and 58 other quantities. We compare the answers of familial sinistrals - individuals who are left-handed themselves or have a left-handed close blood-relative - with those of pure familial dextrals - right-handed individuals who reported only having right-handed close blood-relatives. We show that familial sinistrals use rounder numbers than pure familial dextrals in the survey responses. Round numbers in the decimal system are those that are multiples of powers of 10 or of half or a quarter of a power of 10. Roundness is a gradient concept, e.g. 100 is rounder than 50 or 200. We show that very round number like 100 and 1000 are used with 25% greater likelihood by familial sinistrals than by pure familial dextrals, while pure familial dextrals are more likely to use less round numbers such as 25, 60, and 200. We then use Sigurd's (1988, Language in Society) index of the roundness of a number and report that familial sinistrals' responses are significantly rounder on average than those of pure familial dextrals. To explain the difference, we propose that the cognitive effort of using exact numbers is greater for the familial sinistral group because their language and number systems tend to be more distributed over both hemispheres of the brain. Our data support the view that exact and approximate quantities are processed by two separate cognitive systems. Specifically, our behavioral data corroborates the view that the evolutionarily older, approximate number system is present in both hemispheres of the brain, while the exact number system tends to be localized in only one hemisphere. 198 Wed, 31 Dec 1969 16:33:33 -0800 Sauerland, Uli and Gotzner, Nicole Circuitry in 3D chess /entry/199 199 Wed, 31 Dec 1969 16:33:33 -0800 Goucher, Adam The urinal problem /entry/200 200 Wed, 31 Dec 1969 16:33:33 -0800 Kranakis, Evangelos and Krizanc, Danny A Smaller Sleeping Bag For A Baby Snake /entry/201 201 Wed, 31 Dec 1969 16:33:33 -0800 Linusson, Svante and ASTLUND, JW Constructing the Tits ovoid from an elliptic quadric /entry/202 202 Wed, 31 Dec 1969 16:33:33 -0800 Cherowitzo, WE Das 2: 3-Ei-ein praktikables Eimodell /entry/203 203 Wed, 31 Dec 1969 16:33:33 -0800 Möller, H Problems to sharpen the young /entry/204 An annotated translation of Propositiones ad acuendos juvenes, the oldest mathematical problem collection in Latin, attributed to Alcuin of York. 204 Wed, 31 Dec 1969 16:33:33 -0800 Hadley, John and Singmaster, David The Circle-Squaring Problem Decomposed /entry/205 205 Wed, 31 Dec 1969 16:33:33 -0800 Pierce, Pamela and Ramsay, John Review of "Groups" by Georges Papy in New Scientist /entry/206 206 Wed, 31 Dec 1969 16:33:33 -0800 T. H. O'Beirne Zeroless Arithmetic: Representing Integers ONLY using ONE /entry/207 We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the inputs are ones. We also describe efficient algorithms for the random generation of such representations, and use Dynamical Programming to find a shortest possible formula representing any given positive integer. 207 Wed, 31 Dec 1969 16:33:33 -0800 Ghang, EK and Zeilberger, Doron Circular reasoning: who first proved that $C/d$ is a constant? /entry/208 We answer the question: who first proved that $C/d$ is a constant? We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant ($C/d=A/r^{2}$). He stated neither result explicitly, but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid's $\$emph{Elements}; this was the first step toward a rigorous theory of arc length. We also discuss how Archimedes's work coexisted with the 2000-year belief -- championed by scholars from Aristotle to Descartes -- that it is impossible to find the ratio of a curved line to a straight line. 208 Wed, 31 Dec 1969 16:33:33 -0800 Richeson, David Embedding countable groups in 2-generator groups /entry/209 209 Wed, 31 Dec 1969 16:33:33 -0800 Galvin, Fred The Muddy Children : A logic for public announcement /entry/210 210 Wed, 31 Dec 1969 16:33:33 -0800 Hughes, Jesse What are some of the most ridiculous proofs in mathematics? /entry/211 211 Wed, 31 Dec 1969 16:33:33 -0800 Anonymous Markets are efficient if and only if P = NP /entry/212 212 Wed, 31 Dec 1969 16:33:33 -0800 Maymin, PZ Conway's Rational Tangles /entry/213 213 Wed, 31 Dec 1969 16:33:33 -0800 Davis, Tom A note on paradoxical metric spaces /entry/214 214 Wed, 31 Dec 1969 16:33:33 -0800 Deuber, W A and Simonovits, M and Os, V T S Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality /entry/215 A familiar problem in neo-Riemannian theory is that the P, L, and R operations defined as contextual inversions on pitch-class segments do not produce parsimonious voice leading. We incorporate permutations into T/I-PLR-duality to resolve this issue and simultaneously broaden the applicability of this duality. More precisely, we construct the dual group to the permutation group acting on n-tuples with distinct entries, and prove that the dual group to permutations adjoined with a group G of invertible affine maps Z12 -\textgreater Z12 is the internal direct product of the dual to permutations and the dual to G. Musical examples include Liszt, R. W. Venezia, S. 201 and Schoenberg, String Quartet Number 1, Opus 7. We also prove that the Fiore--Noll construction of the dual group in the finite case works, and clarify the relationship of permutations with the RICH transformation. 215 Wed, 31 Dec 1969 16:33:33 -0800 Fiore, Thomas M. and Noll, Thomas and Satyendra, Ramon Delay can stabilize: Love affairs dynamics /entry/216 We discuss two models of interpersonal interactions with delay. The first model is linear, and allows the presentation of a rigorous mathematical analysis of stability, while the second is nonlinear and a typical local stability analysis is thus performed. The linear model is a direct extension of the classic Strogatz model. On the other hand, as interpersonal relations are nonlinear dynamical processes, the nonlinear model should better reflect real interactions. Both models involve immediate reaction on partner's state and a correction of the reaction after some time. The models we discuss belong to the class of two-variable systems with one delay for which appropriate delay stabilizes an unstable steady state. We formulate a theorem and prove that stabilization takes place in our case. We conclude that considerable (meaning large enough, but not too large) values of time delay involved in the model can stabilize love affairs dynamics. 216 Wed, 31 Dec 1969 16:33:33 -0800 Bielczyk, Natalia and Bodnar, Marek and Foryś, Urszula Algorithmic self-assembly of DNA Sierpinski triangles. /entry/217 Algorithms and information, fundamental to technological and biological organization, are also an essential aspect of many elementary physical phenomena, such as molecular self-assembly. Here we report the molecular realization, using two-dimensional self-assembly of DNA tiles, of a cellular automaton whose update rule computes the binary function XOR and thus fabricates a fractal pattern--a Sierpinski triangle--as it grows. To achieve this, abstract tiles were translated into DNA tiles based on double-crossover motifs. Serving as input for the computation, long single-stranded DNA molecules were used to nucleate growth of tiles into algorithmic crystals. For both of two independent molecular realizations, atomic force microscopy revealed recognizable Sierpinski triangles containing 100-200 correct tiles. Error rates during assembly appear to range from 1% to 10%. Although imperfect, the growth of Sierpinski triangles demonstrates all the necessary mechanisms for the molecular implementation of arbitrary cellular automata. This shows that engineered DNA self-assembly can be treated as a Turing-universal biomolecular system, capable of implementing any desired algorithm for computation or construction tasks. 217 Wed, 31 Dec 1969 16:33:33 -0800 Rothemund, Paul W K and Papadakis, Nick and Winfree, Erik Invited commentary: the perils of birth weight--a lesson from directed acyclic graphs. /entry/218 The strong association of birth weight with infant mortality is complicated by a paradoxical finding: Small babies in high-risk populations usually have lower risk than small babies in low-risk populations. In this issue of the Journal, Hernández-Díaz et al. (Am J Epidemiol 2006;164:1115-20) address this "birth weight paradox" using directed acyclic graphs (DAGs). They conclude that the paradox is the result of bias created by adjustment for a factor (birth weight) that is affected by the exposure of interest and at the same time shares causes with the outcome (mortality). While this bias has been discussed before, the DAGs presented by Hernández-Díaz et al. provide more firmly grounded criticism. The DAGs demonstrate (as do many other examples) that seemingly reasonable adjustments can distort epidemiologic results. In this commentary, the birth weight paradox is shown to be an illustration of Simpson's Paradox. It is possible for a factor to be protective within every stratum of a variable and yet be damaging overall. Questions remain as to the causal role of birth weight. 218 Wed, 31 Dec 1969 16:33:33 -0800 Wilcox, Allen J The paramagnetic and glass transitions in sudoku /entry/219 We study the statistical mechanics of a model glassy system based on a familiar and popular mathematical puzzle. Sudoku puzzles provide a very rare example of a class of frustrated systems with a unique groundstate without symmetry. Here, the puzzle is recast as thermodynamic system where the number of violated rules defines the energy. We use Monte Carlo simulation to show that the "Sudoku Hamiltonian" exhibits two transitions as a function of temperature, a paramagnetic and a glass transition. Of these, the intermediate condensed phase is the only one which visits the ground state (i.e. it solves the puzzle, though this is not the purpose of the study). Both transitions are associated with an entropy change, paramagnetism measured from the dynamics of the Monte Carlo run, showing a peak in specific heat, while the residual glass entropy is determined by finding multiple instances of the glass by repeated annealing. There are relatively few such simple models for frustrated or glassy systems which exhibit both ordering and glass transitions, sudoku puzzles are unique for the ease with which they can be obtained with the proof of the existence of a unique ground state via the satisfiability of all constraints. Simulations suggest that in the glass phase there is an increase in information entropy with lowering temperature. In fact, we have shown that sudoku have the type of rugged energy landscape with multiple minima which typifies glasses in many physical systems, and this puzzling result is a manifestation of the paradox of the residual glass entropy. These readily-available puzzles can now be used as solvable model Hamiltonian systems for studying the glass transition. 219 Wed, 31 Dec 1969 16:33:32 -0800 Williams, Alex and Ackland, Graeme . J. Figures for "Impossible fractals" /entry/220 220 Wed, 31 Dec 1969 16:33:32 -0800 Cameron Browne Biologically Unavoidable Sequences /entry/221 A biologically unavoidable sequence is an infinite gender sequence which occurs in every gendered, infinite genealogical network satisfying certain tame conditions. We show that every eventually periodic sequence is biologically unavoidable (this generalizes Koenig's Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular automata. 221 Wed, 31 Dec 1969 16:33:32 -0800 Alexander, Samuel How to eat 4/9 of a pizza /entry/222 Given two players alternately picking pieces of a pizza sliced by radial cuts, in such a way that after the first piece is taken every subsequent chosen piece is adjacent to some previously taken piece, we provide a strategy for the starting player to get 4/9 of the pizza. This is best possible and settles a conjecture of Peter Winkler. 222 Wed, 31 Dec 1969 16:33:32 -0800 Knauer, Kolja and Micek, Piotr and Ueckerdt, Torsten A stratification of the space of all $k$-planes in $\mathbb{C}_n$ /entry/223 To each $k \times n$ matrix $\mathrm{M}$ of rank $k$, we associate a juggling pattern of periodicity $n$ with $k$ balls. The juggling pattern actually only depends on the $k$-plane spanned by the rows, so gives a decomposition of the “Grassmannian” of all $k$-planes in $n$-space. There are many connections between the geometry and the juggling. For example, the natural topology on the space of matrices induces a partial order on the space of juggling patterns, which indicates whether one pattern is “more excited” than another. This same decomposition turns out to naturally arise from totally positive geometry, characteristic $p$ geometry, and noncommutative geometry. It also arises by projection from the manifold of full flags in $n$-space, where there is no cyclic symmetry 223 Wed, 31 Dec 1969 16:33:32 -0800 Knutson, A Conway's Wizards /entry/224 I present and discuss a puzzle about wizards invented by John H. Conway. 224 Wed, 31 Dec 1969 16:33:32 -0800 Khovanova, Tanya Picture-Hanging Puzzles /entry/225 We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions. 225 Wed, 31 Dec 1969 16:33:32 -0800 Demaine, Erik D. and Demaine, Martin L. and Minsky, Yair N. and Mitchell, Joseph S. B. and Rivest, Ronald L. and Patrascu, Mihai Papy's Minicomputer /entry/226 226 Wed, 31 Dec 1969 16:33:32 -0800 Papy, F The lost squares of Dr. Franklin: Ben Franklin's missing squares and the secret of the magic circle /entry/227 227 Wed, 31 Dec 1969 16:33:32 -0800 Pasles, PC On sphere-filling ropes /entry/228 What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seamlines of a tennis ball, others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells. 228 Wed, 31 Dec 1969 16:33:32 -0800 Gerlach, Henryk and von der Mosel, Heiko Algebraic theory of Penrose's non-periodic tilings of the plane /entry/229 229 Wed, 31 Dec 1969 16:33:32 -0800 Bruijn, NG De Earliest Uses of Symbols of Calculus /entry/230 230 Wed, 31 Dec 1969 16:33:32 -0800 Jeff Miller The topology of the minimal regular cover of the Archimedean tessellations /entry/231 In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps. 231 Wed, 31 Dec 1969 16:33:32 -0800 Coulbois, Thierry and Pellicer, Daniel and Raggi, Miguel and Ramírez, Camilo and Valdez, Ferrán Twin Towers of Hanoi /entry/232 In the Twin Towers of Hanoi version of the well known Towers of Hanoi Problem there are two coupled sets of pegs. In each move, one chooses a pair of pegs in one of the sets and performs the only possible legal transfer of a disk between the chosen pegs (the smallest disk from one of the pegs is moved to the other peg), but also, simultaneously, between the corresponding pair of pegs in the coupled set (thus the same sequence of moves is always used in both sets). We provide upper and lower bounds on the length of the optimal solutions to problems of the following type. Given an initial and a final position of N disks in each of the coupled sets, what is the smallest number of moves needed to simultaneously obtain the final position from the initial one in each set? Our analysis is based on the use of a group, called Hanoi Towers group, of rooted ternary tree automorphisms, which models the original problem in such a way that the configurations on N disks are the vertices at level N of the tree and the action of the generators of the group represents the three possible moves between the three pegs. The twin version of the problem is analyzed by considering the action of Hanoi Towers group on pairs of vertices. 232 Wed, 31 Dec 1969 16:33:32 -0800 Sunic, Zoran Online Dating Recommender Systems: The Split-complex Number Approach /entry/233 233 Wed, 31 Dec 1969 16:33:32 -0800 Jérôme Kunegis Magic: the Gathering is Turing Complete /entry/234 We always knew Magic: the Gathering was a complex game. But now it's proven: you could assemble a computer out of Magic cards. 234 Wed, 31 Dec 1969 16:33:32 -0800 Alex Churchill Modified Pascal Triangle and Pascal Surfaces /entry/235 235 Wed, 31 Dec 1969 16:33:32 -0800 David Alvo Beastly Numbers /entry/236 It seems unlikely that two computers, designed by different people 1800 miles apart, would be upset in the same way by the same two floating-point numbers 65535... and 4294967295... , but it has happened. 236 Wed, 31 Dec 1969 16:33:32 -0800 Kahan, W How Java's floating-point hurts everyone everywhere /entry/237 237 Wed, 31 Dec 1969 16:33:32 -0800 Kahan, W and Darcy, JD A Hamiltonian circuit for Rubik's Cube /entry/238 At last, the Hamiltonian circuit problem for Rubik's Cube has a solution! To be a little more mathematically precise, a Hamiltonian circuit of the quarter-turn metric Cayley graph for the Rubik's Cube group has been found. 238 Wed, 31 Dec 1969 16:33:32 -0800 cuBerBruce How far can Tarzan jump? /entry/239 The tree-based rope swing is a popular recreation facility, often installed in outdoor areas, giving pleasure to thrill-seekers. In the setting, one drops down from a high platform, hanging from a rope, then swings at a great speed like "Tarzan", and finally jumps ahead to land on the ground. The question now arises: How far can Tarzan jump by the swing? In this article, I present an introductory analysis of the Tarzan swing mechanics, a big pendulum-like swing with Tarzan himself attached as weight. The analysis enables determination of how farther forward Tarzan can jump using a given swing apparatus. The discussion is based on elementary mechanics and, therefore, expected to provide rich opportunities for investigations using analytic and numerical methods. 239 Wed, 31 Dec 1969 16:33:32 -0800 Shima, Hiroyuki Mastermind is NP-Complete /entry/240 In this paper we show that the Mastermind Satisfiability Problem (MSP) is NP-complete. The Mastermind is a popular game which can be turned into a logical puzzle called Mastermind Satisfiability Problem in a similar spirit to the Minesweeper puzzle. By proving that MSP is NP-complete, we reveal its intrinsic computational property that makes it challenging and interesting. This serves as an addition to our knowledge about a host of other puzzles, such as Minesweeper, Mah-Jongg, and the 15-puzzle. 240 Wed, 31 Dec 1969 16:33:32 -0800 Stuckman, Jeff and Zhang, Guo-Qiang VIP-club phenomenon: emergence of elites and masterminds in social networks /entry/241 Hubs, or vertices with large degrees, play massive roles in, for example, epidemic dynamics, innovation diffusion, and synchronization on networks. However, costs of owning edges can motivate agents to decrease their degrees and avoid becoming hubs, whereas they would somehow like to keep access to a major part of the network. By analyzing a model and tennis players' partnership networks, we show that combination of vertex fitness and homophily yields a VIP club made of elite vertices that are influential but not easily accessed from the majority. Intentionally formed VIP members can even serve as masterminds, which manipulate hubs to control the entire network without exposing themselves to a large mass. From conventional viewpoints based on network topology and edge direction, elites are not distinguished from many other vertices. Understanding network data is far from sufficient; individualistic factors greatly affect network structure and functions per se. 241 Wed, 31 Dec 1969 16:33:32 -0800 Masuda, Naoki and Konno, Norio The Canonical Basis of $\dot{\mathbf{U}}$ for Type $A_{2}$ /entry/242 The modified quantized enveloping algebra has a remarkable basis, called the canonical basis, which was introduced by Lusztig. In this paper, all these monomial elements of the canonical basis for type $A_{2}$ are determined and we also give a conjecture about all polynomial elements of the canonical basis. 242 Wed, 31 Dec 1969 16:33:32 -0800 Cui, Weideng The Fastest and Shortest Algorithm for All Well-Defined Problems /entry/243 243 Wed, 31 Dec 1969 16:33:32 -0800 Hutter, Marcus Carcassonne and multivariate calculus /entry/244 244 Wed, 31 Dec 1969 16:33:32 -0800 Douglas Weathers A New Rose : The First Simple Symmetric 11-Venn Diagram /entry/245 A symmetric Venn diagram is one that is invariant under rotation, up to a relabeling of curves. A simple Venn diagram is one in which at most two curves intersect at any point. In this paper we introduce a new property of Venn diagrams called crosscut symmetry, which is related to dihedral symmetry. Utilizing a computer search restricted to crosscut symmetry we found many simple symmetric Venn diagrams with 11 curves. This answers an existence question that has been open since the 1960's. The first such diagram that was discovered is shown here. 245 Wed, 31 Dec 1969 16:33:32 -0800 Mamakani, Khalegh and Ruskey, Frank The usefulness of useless knowledge /entry/246 246 Wed, 31 Dec 1969 16:33:32 -0800 Flexner, Abraham Seven Staggering Sequences /entry/247 When the Handbook of Integer Sequences came out in 1973, Philip Morrison gave it an enthusiastic review in the Scientific American and Martin Gardner was kind enough to say in his Mathematical Games column for July 1974 that "every recreational mathematician should buy a copy forthwith." That book contained 2372 sequences. Today the On-Line Encyclopedia of Integer Sequences (or OEIS) contains 117000 sequences. The following are seven that I find especially interesting. Many of them quite literally stagger. The sequences will be labeled with their numbers (such as A064413) in the OEIS. Much more information about them can be found there and in the references cited. 247 Wed, 31 Dec 1969 16:33:32 -0800 Sloane, N J A The top ten prime numbers /entry/248 248 Wed, 31 Dec 1969 16:33:32 -0800 Dubner, H Train Sets /entry/249 249 Wed, 31 Dec 1969 16:33:32 -0800 Chalcraft, Adam and Greene, Michael Equilibrium solution to the lowest unique positive integer game /entry/250 We address the equilibrium concept of a reverse auction game so that no one can enhance the individual payoff by a unilateral change when all the others follow a certain strategy. In this approach the combinatorial possibilities to consider become very much involved even for a small number of players, which has hindered a precise analysis in previous works. We here present a systematic way to reach the solution for a general number of players, and show that this game is an example of conflict between the group and the individual interests. 250 Wed, 31 Dec 1969 16:33:32 -0800 Baek, Seung Ki and Bernhardsson, Sebastian The wobbly garden table /entry/251 251 Wed, 31 Dec 1969 16:33:32 -0800 Kraft, Hanspeter A cohomological viewpoint on elementary school arithmetic /entry/252 252 Wed, 31 Dec 1969 16:33:32 -0800 Isaksen, DC On distributions computable by random walks on graphs /entry/253 253 Wed, 31 Dec 1969 16:33:32 -0800 Kindler, G To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction /entry/254 The lambda calculus, and the closely related theory of combinators, are important in the foundations of mathematics, logic and computer science. This paper provides an informal and entertaining introduction by means of an animated graphical notation. 254 Wed, 31 Dec 1969 16:33:32 -0800 David C Keenan Topology Explains Why Automobile Sunshades Fold Oddly /entry/255 We use braids and linking number to explain why automobile shades fold into an odd number of loops. 255 Wed, 31 Dec 1969 16:33:32 -0800 Feist, Curtis and Naimi, Ramin On an error in the star puzzle by Henry E. Dudeney /entry/256 We found a solution of the star puzzle (a path on a chessboard from c5 to d4 in 14 straight strokes) in 14 queen moves, which has been claimed by the author as impossible. 256 Wed, 31 Dec 1969 16:33:32 -0800 Ravsky, Alex How to recognise a 4-ball when you see one /entry/257 We apply the method of filling with holomorphic discs to a 4-dimensional symplectic cobordism with the standard contact 3-sphere as a convex boundary component. We establish the following dichotomy: either the cobordism is diffeomorphic to a ball, or there is a periodic Reeb orbit of quantifiably short period in the concave boundary of the cobordism. This allows us to give a unified treatment of various results concerning Reeb dynamics on contact 3-manifolds, symplectic fillability, the topology of symplectic cobordisms, symplectic non-squeezing, and the non-existence of exact Lagrangian surfaces in standard symplectic 4-space. 257 Wed, 31 Dec 1969 16:33:32 -0800 Geiges, Hansjörg and Zehmisch, Kai G2 and the Rolling Ball /entry/258 Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2: it acts as the symmetries of a 'spinorial ball rolling on a projective plane', again when the ratio of radii is 1:3. We explain this ratio in simple terms using the incidence geometry of G2, and show how a form of geometric quantization applied to this system gives the imaginary split octonions. 258 Wed, 31 Dec 1969 16:33:32 -0800 Baez, John C and Huerta, John Lectures on lost mathematics /entry/259 259 Wed, 31 Dec 1969 16:33:32 -0800 Branko Grünbaum Estimating the Effect of the Red Card in Soccer /entry/260 We study the effect of the red card in a soccer game. A red card is given by a referee to signify that a player has been sent off following a serious misconduct. The player who has been sent off must leave the game immediately and cannot be replaced during the game. His team must continue the game with one player fewer. We estimate the effect of the red card from betting data on the FIFA World Cup 2006 and Euro 2008, showing that the scoring intensity of the penalized team drops significantly, while the scoring intensity of the opposing team increases slightly. We show that a red card typically leads to a smaller number of goals scored during the game when a stronger team is penalized, but it can lead to an increased number of goals when a weaker team is punished. We also show when it is better to commit a red card offense in exchange for the prevention of a goal opportunity. 260 Wed, 31 Dec 1969 16:33:32 -0800 Vecer, Jan and Kopriva, Frantisek A categorical foundation for Bayesian probability /entry/261 Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a prior probability measure $P_H$ on $H$ and a sampling distribution $\mcS:H \rightarrow D$, there is a corresponding inference map $\mcI:D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\mu:1 \rightarrow D$, a posterior probability $\hat{P_H}=\mcI \circ \mu$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $\mcI$ and the process repeats with each additional measurement. The main result shows that the assumption of Polish spaces to obtain regular conditional probabilities is not necessary---countably generated spaces suffice. This less stringent condition then allows for non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as non finite) spaces, and also provides for a common framework for decision theory and Bayesian probability. 261 Wed, 31 Dec 1969 16:33:32 -0800 Culbertson, Jared and Sturtz, Kirk Cardinal arithmetic for skeptics /entry/262 When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with "consistency" rather than "truth" may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of $2^{\aleph_0}$, cannot be settled on the basis of the usual axioms of set theory (ZFC). Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic. 262 Wed, 31 Dec 1969 16:33:32 -0800 Shelah, Saharon Survey on fusible numbers /entry/263 We point out that the recursive formula that appears in Erickson's presentation "Fusible Numbers" is incorrect, and pose an alternate conjecture about the structure of fusible numbers. Although we are unable to solve the conjecture, we succeed in establishing some basic properties of fusible numbers. We suggest some possible approaches to the conjecture, and list further problems in the final chapter. 263 Wed, 31 Dec 1969 16:33:32 -0800 Xu, Junyan A mathematician's survival guide /entry/264 264 Wed, 31 Dec 1969 16:33:32 -0800 Casazza, Peter G Calculus Made Easy /entry/265 Being a very simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the DIFFERENTIAL CALCULUS and the INTEGRAL CALCULUS 265 Wed, 31 Dec 1969 16:33:32 -0800 Thompson, Silvanus P Long finite sequences /entry/266 Let k be a positive integer. There is a longest finite sequence x1,...,xn in k letters in which no consecutive block xi,...,x2i is a subsequence of any other consecutive block xj,...,x2j. Let n(k) be this longest length. We prove that n(1) = 3, n(2) = 11, and n(3) is incomprehensibly large. We give a lower bound for n(3) in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for n(k). We view n(3) as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for n(3). 266 Wed, 31 Dec 1969 16:33:32 -0800 Friedman, Harvey M Computer analysis of Sprouts with nimbers /entry/267 Sprouts is a two-player topological game, invented in 1967 in the University of Cambridge by John Conway and Michael Paterson. The game starts with p spots, and ends in at most 3p-1 moves. The first player who cannot play loses. The complexity of the p-spot game is very high, so that the best hand-checked proof only shows who the winner is for the 7-spot game, and the best previous computer analysis reached p=11. We have written a computer program, using mainly two new ideas. The nimber (also known as Sprague-Grundy number) allows us to compute separately independent subgames; and when the exploration of a part of the game tree seems to be too difficult, we can manually force the program to search elsewhere. Thanks to these improvements, we reached up to p=32. The outcome of the 33-spot game is still unknown, but the biggest computed value is the 47-spot game ! All the computed values support the Sprouts conjecture: the first player has a winning strategy if and only if p is 3, 4 or 5 modulo 6. We have also used a check algorithm to reduce the number of positions needed to prove which player is the winner. It is now possible to hand-check all the games until p=11 in a reasonable amount of time. 267 Wed, 31 Dec 1969 16:33:32 -0800 Lemoine, Julien and Viennot, Simon Nim multiplication /entry/268 268 Wed, 31 Dec 1969 16:33:32 -0800 H. W. Lenstra, Jr. Theory and History of Geometric Models /entry/269 269 Wed, 31 Dec 1969 16:33:32 -0800 Polo-blanco, Irene Robust Soldier Crab Ball Gate /entry/270 Based on the field observation of soldier crabs, we previously proposed a model for a swarm of soldier crabs. Here, we describe the interaction of coherent swarms in the simulation model, which is implemented in a logical gate. Because a swarm is generated by inherent perturbation, a swarm can be generated and maintained under highly perturbed conditions. Thus, the model reveals a robust logical gate rather than stable one. In addition, we show that the logical gate of swarms is also implemented by real soldier crabs (Mictyris guinotae). 270 Wed, 31 Dec 1969 16:33:32 -0800 Gunji, YP and Nishiyama, Y Statistical Modeling of Gang Violence in Los Angeles /entry/271 271 Wed, 31 Dec 1969 16:33:32 -0800 Fathauer, Chris High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams /entry/272 This volume consists of a selection of papers based on presentations made at the international conference on number theory held in honor of Hugh Williams' sixtieth birthday. The papers address topics in the areas of computational and explicit number theory and its applications. The material is suitable for graduate students and researchers interested in number theory. 272 Wed, 31 Dec 1969 16:33:32 -0800 Williams, Hugh C. and Poorten, A. J. Van Der and Stein, Andreas Light reflecting off Christmas-tree balls /entry/273 'Twas the night before Christmas and under the tree Was a heap of new balls, stacked tight as can be. The balls so gleaming, they reflect all light rays, Which bounce in the stack every which way. When, what to my wondering mind does occur: A question of interest; I hope you concur! From each point outside, I wondered if light Could reach deep inside through gaps so tight? 273 Wed, 31 Dec 1969 16:33:32 -0800 Joseph O'Rourke Cellular automata in the hyperbolic plane: proposal for a new environment /entry/274 274 Wed, 31 Dec 1969 16:33:32 -0800 Chelghoum, Kamel and Margenstern, Maurice and Martin, Beno\^it Cake Cutting Mechanisms /entry/275 We examine the history of cake cutting mechanisms and discuss the efficiency of their allocations. In the case of piecewise uniform preferences, we define a game that in the presence of strategic agents has equilibria that are not dominated by the allocations of any mechanism. We identify that the equilibria of this game coincide with the allocations of an existing cake cutting mechanism. 275 Wed, 31 Dec 1969 16:33:32 -0800 Ianovski, Egor Navigating Hyperbolic Space with Fibonacci Trees /entry/276 276 Wed, 31 Dec 1969 16:33:32 -0800 moniker The Euler spiral: a mathematical history /entry/277 The beautiful Euler spiral, defined by the linear relationship between curvature and arclength, was first proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, first by Augustin Fresnel to compute diffraction of light through a slit, and again by Arthur Talbot to produce an ideal shape for a railway transition curve connecting a straight section with a section of given curvature. Though it has gathered many names throughout its history, the curve retains its aesthetic and mathematical beauty as Euler had clearly visualized. Its equation is related to the Gamma function, the Gauss error function (erf), and is a special case of the confluent hypergeometric function. 277 Wed, 31 Dec 1969 16:33:32 -0800 Levien, Raph Undecidable problems: a sampler /entry/278 After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 278 Wed, 31 Dec 1969 16:33:32 -0800 Poonen, Bjorn The mate-in-n problem of infinite chess is decidable /entry/279 Infinite chess is chess played on an infinite edgeless chessboard. The familiar chess pieces move about according to their usual chess rules, and each player strives to place the opposing king into checkmate. The mate-in-n problem of infinite chess is the problem of determining whether a designated player can force a win from a given finite position in at most n moves. A naive formulation of this problem leads to assertions of high arithmetic complexity with 2n alternating quantifiers---there is a move for white, such that for every black reply, there is a counter-move for white, and so on. In such a formulation, the problem does not appear to be decidable; and one cannot expect to search an infinitely branching game tree even to finite depth. Nevertheless, the main theorem of this article, confirming a conjecture of the first author and C. D. A. Evans, establishes that the mate-in-n problem of infinite chess is computably decidable, uniformly in the position and in n. Furthermore, there is a computable strategy for optimal play from such mate-in-n positions. The proof proceeds by showing that the mate-in-n problem is expressible in what we call the first-order structure of chess, which we prove (in the relevant fragment) is an automatic structure, whose theory is therefore decidable. Indeed, it is definable in Presburger arithmetic. Unfortunately, this resolution of the mate-in-n problem does not appear to settle the decidability of the more general winning-position problem, the problem of determining whether a designated player has a winning strategy from a given position, since a position may admit a winning strategy without any bound on the number of moves required. This issue is connected with transfinite game values in infinite chess, and the exact value of the omega one of chess is not known. 279 Wed, 31 Dec 1969 16:33:32 -0800 Brumleve, Dan and Hamkins, Joel David and Schlicht, Philipp Statistical Laws Governing Fluctuations in Word Use from Word Birth to Word Death /entry/280 We analyze the dynamic properties of 10^7 words recorded in English, Spanish and Hebrew over the period 1800--2008 in order to gain insight into the coevolution of language and culture. We report language independent patterns useful as benchmarks for theoretical models of language evolution. A significantly decreasing (increasing) trend in the birth (death) rate of words indicates a recent shift in the selection laws governing word use. For new words, we observe a peak in the growth-rate fluctuations around 40 years after introduction, consistent with the typical entry time into standard dictionaries and the human generational timescale. Pronounced changes in the dynamics of language during periods of war shows that word correlations, occurring across time and between words, are largely influenced by coevolutionary social, technological, and political factors. We quantify cultural memory by analyzing the long-term correlations in the use of individual words using detrended fluctuation analysis. 280 Wed, 31 Dec 1969 16:33:32 -0800 Petersen, Alexander M and Tenenbaum, Joel and Havlin, Shlomo and Stanley, H Eugene The bitangent sphere problem /entry/281 281 Wed, 31 Dec 1969 16:33:32 -0800 Giblin, PJ Gaussian prime spirals /entry/282 Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer, moving initially $\pm$ in the horizontal or vertical directions. When it hits a Gaussian prime, it turns left $90^{\circ}$. 282 Wed, 31 Dec 1969 16:33:32 -0800 Joseph O'Rourke Topic-based vector space model /entry/283 283 Wed, 31 Dec 1969 16:33:32 -0800 Becker, Jörg A New Approximation to $\pi$ (Conclusion) /entry/284 284 Wed, 31 Dec 1969 16:33:32 -0800 Ferguson, D. F. and Wrench, John W Doc, What Are My Chances? /entry/285 285 Wed, 31 Dec 1969 16:33:32 -0800 Marasco, Joe and Doerfler, Ron and Roschier, Leif Random walks reaching against all odds the other side of the quarter plane /entry/286 For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state $(i_0,j_0)$, we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive an exact expression for the probability of this event, and derive an asymptotic expression for the case when $i_0$ becomes large, a situation in which the event becomes highly unlikely. The exact expression follows from the solution of a boundary value problem and is in terms of an integral that involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model and the asymmetric exclusion process. 286 Wed, 31 Dec 1969 16:33:32 -0800 van Leeuwaarden, Johan S. H. and Raschel, Kilian Quotients Homophones des Groupes Libres Homophonic Quotients of Free Groups /entry/287 287 Wed, 31 Dec 1969 16:33:32 -0800 Washington, Lawrence and Zagier, Don Poe, E.: Near A Raven /entry/288 At the time of its writing in 1995, this composition in Standard Pilish, a retelling of Edgar Allan Poe's "The Raven", was one of the longest texts ever written using the π constraint, in which the number of letters in each successive word "spells out" the digits of π (740 digits in this example). For length this poem was subsequently outdone by the nearly-4000-digit Cadaeic Cadenza, whose first section is just Near A Raven with the first three words altered, but since this version is fairly well-known by itself (for example, it was reprinted in Berggren, Borwein and Borwein's "Pi: A Source Book"), we have decided to give it its own web page. 288 Wed, 31 Dec 1969 16:33:32 -0800 Mike Keith Barcodes: the persistent topology of data /entry/289 289 Wed, 31 Dec 1969 16:33:32 -0800 Ghrist, Robert Further evidence for addition and numerical competence by a Grey parrot (Psittacus erithacus) /entry/290 A Grey parrot ( Psittacus erithacus ), able to quantify sets of eight or fewer items (including heterogeneous subsets), to sum two sequentially presented sets of 0–6 items (up to 6), and to identify and serially order Arabic numerals (1–8), all by using English labels (Pepperberg in J Comp Psychol 108:36–44, 1994 ; J Comp Psychol 120:1–11, 2006a ; J Comp Psychol 120:205–216, 2006b ; Pepperberg and Carey submitted), was tested on addition of two Arabic numerals or three sequentially presented collections (e.g., of variously sized jelly beans or nuts). He was, without explicit training and in the absence of the previously viewed addends, asked, “How many total?” and required to answer with a vocal English number label. In a few trials on the Arabic numeral addition, he was also shown variously colored Arabic numerals while the addends were hidden and asked “What color number (is the) total?” Although his death precluded testing on all possible arrays, his accuracy was statistically significant and suggested addition abilities comparable with those of nonhuman primates. 290 Wed, 31 Dec 1969 16:33:32 -0800 Pepperberg, Irene M. Passage to the limit in Proposition I, Book I of Newton's Principia /entry/291 291 Wed, 31 Dec 1969 16:33:32 -0800 Erlichson, Herman Orange Peels and Fresnel Integrals /entry/292 There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife. 292 Wed, 31 Dec 1969 16:33:32 -0800 Bartholdi, Laurent and Henriques, André G. Tropical Mathematics /entry/293 These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the ``tropical approach'' in mathematics, which has gotten a lot of attention recently in combinatorics, algebraic geometry and related fields. It offers an an elementary introduction to this subject, touching upon Arithmetic, Polynomials, Curves, Phylogenetics and Linear Spaces. Each section ends with a suggestion for further research. The bibliography contains numerousreferences for further reading in this field. 293 Wed, 31 Dec 1969 16:33:32 -0800 Speyer, David and Sturmfels, Bernd Pantologia. A new (cabinet) cyclopædia, by J.M. Good, O. Gregory, and N. Bosworth assisted by other gentlemen of eminence /entry/294 294 Wed, 31 Dec 1969 16:33:32 -0800 Good, John Mason and Gregory, Olinthus Gilbert Fractions without Quotients: Arithmetic of Repeating Decimals /entry/295 295 Wed, 31 Dec 1969 16:33:32 -0800 Plagge, Richard Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later /entry/296 296 Wed, 31 Dec 1969 16:33:32 -0800 Moler, Cleve and Van Loan, C. Good stories, pity they're not true /entry/297 297 Wed, 31 Dec 1969 16:33:32 -0800 Devlin, Keith Fibonacci determinants-a combinatorial approach /entry/298 298 Wed, 31 Dec 1969 16:33:32 -0800 Benjamin, A.T. and Cameron, N.T. and Quinn, J.J. Cryptographic and Physical Zero-Knowledge Proof Systems for Solutions of Sudoku Puzzles /entry/299 299 Wed, 31 Dec 1969 16:33:32 -0800 Gradwohl, Ronen and Naor, M. and Pinkas, Benny and Rothblum, G. Benjamin Peirce and the Howland will /entry/300 300 Wed, 31 Dec 1969 16:33:32 -0800 Meier, Paul and Zabell, Sandy Mapping an unfriendly subway system /entry/301 We consider a class of highly dynamic networks modelled on an urban subway system. We examine the problem of creating a map of such a subway in less than ideal conditions, where the local residents are not enthusiastic about the process and there is a limited ability to communicate amongst the mappers. More precisely, we study the problem of a team of asynchronous computational entities (the mapping agents) determining the location of black holes in a highly dynamic graph, whose edges are defined by the asynchronous movements ofmobile entities (the subway carriers). We present and analyze a solution protocol. The algorithm solves the problem with the minimum number of agents possible. We also establish lower bounds on the number of carrier moves in the worst case, showing that our protocol is also move-optimal. 301 Wed, 31 Dec 1969 16:33:32 -0800 Flocchini, Paola and Kellett, Matthew and Mason, P. In retrospect: On the Six-Cornered Snowflake /entry/302 302 Wed, 31 Dec 1969 16:33:32 -0800 Ball, Philip Tropical Arithmetic and Tropical Matrix Algebra /entry/303 This paper introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e. summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular. 303 Wed, 31 Dec 1969 16:33:32 -0800 Izhakian, Zur Gerrymandering and Convexity /entry/304 304 Wed, 31 Dec 1969 16:33:32 -0800 Hodge, Jonathan K. and Marshall, Emily and Patterson, Geoff Continued fractions constructed from prime numbers /entry/305 We give 50 digits values of the simple continued fractions whose denominators are formed from a) prime numbers, b) twin primes, c) generalized $d$-twins, d) primes of the form $m^2+n^4$, e)primes of the form $m^2+1$, f) Mersenne primes and g) primorial primes. All these continued fractions belong to the set of measure zero of exceptions to the theorems of Khinchin and Levy. We claim that all these continued fractions are transcendental numbers. Next we propose the conjecture which indicates the way to deduce the transcendence of some continued fractions from transcendence of another ones. 305 Wed, 31 Dec 1969 16:33:32 -0800 Wolf, Marek Compositional Reasoning Using Intervals and Time Reversal /entry/306 306 Wed, 31 Dec 1969 16:33:32 -0800 Moszkowski, Ben The hardness of the Lemmings game, or Oh no, more NP-completeness proofs /entry/307 307 Wed, 31 Dec 1969 16:33:32 -0800 Cormode, Graham Gaming is a hard job, but someone has to do it! /entry/308 We establish some general schemes relating the computational complexity of a video game to the presence of certain common elements or mechanics, such as destroyable paths, collecting items, doors activated by switches or pressure plates, etc.. Then we apply such "metatheorems" to several video games published between 1980 and 1998, including Pac-Man, Tron, Lode Runner, Boulder Dash, Deflektor, Mindbender, Pipe Mania, Skweek, Prince of Persia, Lemmings, Doom, Puzzle Bobble 3, and Starcraft. We obtain both new results, and improvements or alternative proofs of previously known results. 308 Wed, 31 Dec 1969 16:33:32 -0800 Viglietta, Giovanni London Calling Philosophy and Engineering: WPE 2008 /entry/309 309 Wed, 31 Dec 1969 16:33:32 -0800 Glen Miller The Snowblower Problem /entry/310 We introduce the snowblower problem (SBP), a new optimization problem that is closely related to milling problems and to some material-handling problems. The objective in the SBP is to compute a short tour for the snowblower to follow to remove all the snow from a domain (driveway, sidewalk, etc.). When a snowblower passes over each region along the tour, it displaces snow into a nearby region. The constraint is that if the snow is piled too high, then the snowblower cannot clear the pile. We give an algorithmic study of the SBP. We show that in general, the problem is NP-complete, and we present polynomial-time approximation algorithms for removing snow under various assumptions about the operation of the snowblower. Most commercially-available snowblowers allow the user to control the direction in which the snow is thrown. We differentiate between the cases in which the snow can be thrown in any direction, in any direction except backwards, and only to the right. For all cases, we give constant-factor approximation algorithms; the constants increase as the throw direction becomes more restricted. Our results are also applicable to robotic vacuuming (or lawnmowing) with bounded capacity dust bin and to some versions of material-handling problems, in which the goal is to rearrange cartons on the floor of a warehouse. 310 Wed, 31 Dec 1969 16:33:32 -0800 Arkin, Esther M. and Bender, Michael A. and Mitchell, Joseph S. B. and Polishchuk, Valentin Scooping the Loop Snooper /entry/311 311 Wed, 31 Dec 1969 16:33:32 -0800 Geoffrey K. Pullum Ethnomathematics as a new research field , illustrated by studies of mathematical ideas in African history /entry/312 312 Wed, 31 Dec 1969 16:33:32 -0800 Gerdes, Paulus Drawings from Angola: living mathematics /entry/313 For children from age 8 to 14."Drawings from Angola" present an introduction to an African story telling tradition. The tales are illustrated with marvelous drawings made in the sand. The book conveys the stories of the stork and the leopard, the hunter and the dog, the rooster and the fox, and others. It explains how to execute the drawings. The reader is invited to draw tortoises, antelopes, lions, and other animals. The activities proposed throughout the book invite the reader to experiment and to explore the 'rhythm' and symmetry of the illustrations. Surprising results will be playfully obtained, such as in arithmetic, a way to calculate quickly the sum of a sequence of odd numbers. Children will live the beautiful mathematics of the Angolan sanddrawings.Answers to the activities are provided.The book can be used both in classrooms and at home. 313 Wed, 31 Dec 1969 16:33:32 -0800 Paulus Gerdes Understanding Monads With JavaScript /entry/314 For the past weeks I've been working hard studying monads. I'm still learning Haskell, and to be honest I thought I knew what monads are all about, but when I wanted to write a little Haskell library, just to sharpen up my skills, I realized that while I understood the way monadic bind (>>=) and return work, I had no understanding of where that state comes from. So, most likely I had no understanding at all. As a result of this I thought I rediscover monads myself using JavaScript. The plan was basically the same as that used when I derived the Y Combinator: start from the initial problem (dealing with explicit immutable state in this case), and work my way up to the solution by applying simple code transformations. 314 Wed, 31 Dec 1969 16:33:32 -0800 Ionuț G. Stan The Collatz Fractal /entry/315 315 Wed, 31 Dec 1969 16:33:32 -0800 Henderson, Xander Analysis of Casino Shelf Shuffling Machines /entry/316 Many casinos routinely use mechanical card shuffling machines. We were asked to evaluate a new product, a shelf shuffler. This leads to new probability, new combinatorics, and to some practical advice which was adopted by the manufacturer. The interplay between theory, computing, and real-world application is developed. 316 Wed, 31 Dec 1969 16:33:32 -0800 Diaconis, Persi and Fulman, Jason and Holmes, Susan A zero-knowledge Poker protocol that achieves confidentiality of the players' strategy or How to achieve an electronic Poker face /entry/317 317 Wed, 31 Dec 1969 16:33:32 -0800 Crépeau, C. A Generalized Fibonacci LSB Data Hiding Technique /entry/318 318 Wed, 31 Dec 1969 16:33:32 -0800 Battisti, F and Carli, M and Neri, A and Egiaziarian, K Computer evolution of buildable objects /entry/319 319 Wed, 31 Dec 1969 16:33:32 -0800 Funes, Pablo and Pollack, Jordan Random Walks on Finite Groups /entry/320 Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as the cut-off phenomenon asserts that this often happens abruptly so that it really makes sense to talk about “the mixing time”. Random walks on finite groups generalize card shuffling models by replacing the symmetric group by other finite groups. One then would like to understand how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups. It turns out that this is an extremely rich problem which is very far to be understood. Techniques from a great variety of different fields – Probability, Algebra, Representation Theory, Functional Analysis, Geometry, Combinatorics – have been used to attack special instances of this problem. This article gives a general overview of this area of research. 320 Wed, 31 Dec 1969 16:33:32 -0800 Saloff-coste, Laurent Complexity of Langton's ant /entry/321 321 Wed, 31 Dec 1969 16:33:32 -0800 Gajardo, A and Moreira, A and Goles, E Three-dimensional finite point groups and the symmetry of beaded beads /entry/322 322 Wed, 31 Dec 1969 16:33:32 -0800 Fisher, GL and Mellor, B. On badly approximable numbers and certain games /entry/323 323 Wed, 31 Dec 1969 16:33:32 -0800 Schmidt, WM Carrots for dessert /entry/324 Carrots for dessert is the title of a section of the paper `On polynomial-like mappings' by Douady and Hubbard. In that section the authors define a notion of dyadic carrot fields of the Mandelbrot set M and more generally for Mandelbrot like families. They remark that such carrots are small when the dyadic denominator is large, but they do not even try to prove a precise such statement. In this paper we formulate and prove a precise statement of asymptotic shrinking of dyadic Carrot-fields around M. The same proof carries readily over to show that the dyadic decorations of copies M' of the Mandelbrot set M inside M and inside the parabolic Mandelbrot set shrink to points when the denominator diverge to infinity. 324 Wed, 31 Dec 1969 16:33:32 -0800 Petersen, Carsten Lunde and Roesch, Pascale How to Gamble If You're In a Hurry /entry/325 The beautiful theory of statistical gambling, started by Dubins and Savage (for subfair games) and continued by Kelly and Breiman (for superfair games) has mostly been studied under the unrealistic assumption that we live in a continuous world, that money is indefinitely divisible, and that our life is indefinitely long. Here we study these fascinating problems from a purely discrete, finitistic, and computational, viewpoint, using Both Symbol-Crunching and Number-Crunching (and simulation just for checking purposes). 325 Wed, 31 Dec 1969 16:33:31 -0800 Ekhad, Shalosh B and Georgiadis, Evangelos and Zeilberger, Doron Origami Burrs and Woven Polyhedra /entry/326 326 Wed, 31 Dec 1969 16:33:31 -0800 Lang, Robert J Deobfuscation is in NP /entry/327 327 Wed, 31 Dec 1969 16:33:31 -0800 Appel, Andrew W Designing tie knots by random walks /entry/328 The simplest of conventional tie knots, the four-in-hand, has its origins in late-nineteenth-century England. The Duke of Windsor, as King Edward VIII became after abdicating in 1936, is credited with introducing what is now known as the Windsor knot, from which its smaller derivative, the half-Windsor, evolved. In 1989, the Pratt knot, the first new knot to appear in fifty years, was revealed on the front page of The New York Times. 328 Wed, 31 Dec 1969 16:33:31 -0800 Thomas M. Fink and Yong Mao Tie knots, random walks and topology /entry/329 329 Wed, 31 Dec 1969 16:33:31 -0800 Fink, T and Mao, Y What Are the Odds? /entry/330 Gambling Has No Place in Baseball But Every Move on the Diamond Is Governed by the Laws of Chance--- The Successful Manager Is Successful Just So Far As He Knows and Accepts the Odds 330 Wed, 31 Dec 1969 16:33:31 -0800 Lane, F.C. Asymptotic statistics of the n-sided planar Poisson–Voronoi cell: I. Exact results /entry/331 331 Wed, 31 Dec 1969 16:33:31 -0800 Hilhorst, H.J. Laying train tracks /entry/332 This morning I was playing trains with my son Felix. At the moment he is much more interested in laying the tracks than putting the trains on and moving them around, but he doesn’t tend to get concerned about whether the track closes up to make a loop. 332 Wed, 31 Dec 1969 16:33:31 -0800 Danny Calegari Tetris is Hard, Even to Approximate /entry/333 In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p^(1-epsilon), when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of (2 - epsilon), for any epsilon>0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piece sets. 333 Wed, 31 Dec 1969 16:33:31 -0800 Demaine, Erik D and Hohenberger, Susan and Liben-Nowell, David Remainder Wheels and Group Theory /entry/334 334 Wed, 31 Dec 1969 16:33:31 -0800 Brenton, Lawrence Chalk : Materials and Concepts in Mathematics Chalk in Hand /entry/335 335 Wed, 31 Dec 1969 16:33:31 -0800 Barany, Michael J and Mackenzie, Donald The experimental effectiveness of mathematical proof /entry/336 336 Wed, 31 Dec 1969 16:33:31 -0800 Miquel, Alexandre Scholarly communication in transition: The use of question marks in the titles of scientific articles in medicine, life sciences and physics 1966–2005 /entry/337 The titles of scientific articles have a special significance. We examined nearly 20 million scientific articles and recorded the development of articles with a question mark at the end of their titles over the last 40 years. Our study was confined to the disciplines of physics, life sciences and medicine, where we found a significant increase from 50% to more than 200% in the number of articles with question-mark titles. We looked at the principle functions and structure of the titles of scientific papers, and we assume that marketing aspects are one of the decisive factors behind the growing usage of question-mark titles in scientific articles. 337 Wed, 31 Dec 1969 16:33:31 -0800 Ball, Rafael Baron Munchhausen Redeems Himself : Bounds for a Coin-Weighing Puzzle Background /entry/338 We investigate a coin-weighing puzzle that appeared in the Moscow Math Olympiad in 1991. We generalize the puzzle by varying the number of participating coins, and deduce an upper bound on the number of weighings needed to solve the puzzle that is noticeably better than the trivial upper bound. In particular, we show that logarithmically-many weighings on a balance suffice. 338 Wed, 31 Dec 1969 16:33:31 -0800 Khovanova, Tanya and Lewis, Joel Brewster Cool irrational numbers and their rather cool rational approximations /entry/339 339 Wed, 31 Dec 1969 16:33:31 -0800 Calogero, Francesco A linear programming approach for aircraft boarding strategy /entry/340 340 Wed, 31 Dec 1969 16:33:31 -0800 Bazargan, M The elasto-plastic indentation of a half-space by a rigid sphere /entry/341 341 Wed, 31 Dec 1969 16:33:31 -0800 Hardy, C. and Baronet, C. N. and Tordion, G. V. Gödel's Second Incompleteness Theorem Explained in Words of One Syllable /entry/342 342 Wed, 31 Dec 1969 16:33:31 -0800 Boolos, George Fusible Numbers /entry/343 343 Wed, 31 Dec 1969 16:33:31 -0800 Erickson, Jeff There is no "Uspensky's method" /entry/344 In this paper an attempt is made to correct the misconception of several authors that there exists a method by Upensky (based on Vincent's theorem) for the isolation of the real roots of a polynomial equation with rational coefficients. Despite Uspensky's claim, in the preface of his book, that he invented this method, we show that what Uspensky actually did was to take Vincent's method and double its computing time. Uspensky must not have understood Vincent's method probably because he was not aware of Budan's theorem. In view of the above, it is historically incorrect to attribute Vincent's method to Uspensky. 344 Wed, 31 Dec 1969 16:33:31 -0800 Akritas, AG Mad Abel : A card game for 2 + players /entry/345 345 Wed, 31 Dec 1969 16:33:31 -0800 Mccarthy, Smári Does Quantum Interference exist in Twitter? /entry/346 It becomes more difficult to explain the social information transfer phenomena using the classic models based merely on Shannon Information Theory (SIT) and Classic Probability Theory (CPT), because the transfer process in the social world is rich of semantic and highly contextualized. This paper aims to use twitter data to explore whether the traditional models can interpret information transfer in social networks, and whether quantum-like phenomena can be spotted in social networks. Our main contributions are: (1) SIT and CPT fail to interpret the information transfer occurring in Twitter; and (2) Quantum interference exists in Twitter, and (3) a mathematical model is proposed to elucidate the spotted quantum phenomena. 346 Wed, 31 Dec 1969 16:33:31 -0800 Shuai, Xin and Ding, Ying and Busemeyer, Jerome and Sun, Yuyin and Chen, Shanshan and Tang, Jie Shamos's Catalog of the Real Numbers /entry/347 347 Wed, 31 Dec 1969 16:33:31 -0800 Shamos, Michael Ian Packing circles and spheres on surfaces /entry/348 Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces’ incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circle packings, hexagonal meshes which carry a torsion-free support structure, hybrid tri-hex meshes, and others. We show how triangle meshes can be optimized so as to have the incircle packing property. We explain their relation to conformal geometry and implications on solvability of optimization. The examples we give confirm that this kind of meshes is a rich source of geometric structures relevant to architectural geometry. 348 Wed, 31 Dec 1969 16:33:31 -0800 Alexander Schiftner and Mathias Höbinger and Johannes Wallner and Helmut Pottmann Irrationality from the book /entry/349 349 Wed, 31 Dec 1969 16:33:31 -0800 Miller, Steven J and Montague, David Against Conditionalization /entry/350 350 Wed, 31 Dec 1969 16:33:31 -0800 Bacchus, Fahiem Investigations of Game of Life cellular automata rules on Penrose Tilings : lifetime and ash statistics /entry/351 351 Wed, 31 Dec 1969 16:33:31 -0800 Owens, Nick and Stepney, Susan Doubly-, triply-, quadruply- and quintuply-innervated crustacean muscles /entry/352 352 Wed, 31 Dec 1969 16:33:31 -0800 van Harreveld, A. Gödel's incompleteness theorem /entry/353 353 Wed, 31 Dec 1969 16:33:31 -0800 Uspensky, V Penrose's Godelian argument /entry/354 354 Wed, 31 Dec 1969 16:33:31 -0800 Feferman, Solomon Deriving Uniform Polyhedra with Wythoff's Construction /entry/355 355 Wed, 31 Dec 1969 16:33:31 -0800 Romano, Don Testing Petri Nets for Mobile Robots Using Gröbner Bases /entry/356 356 Wed, 31 Dec 1969 16:33:31 -0800 Chandler, Angie and Heyworth, Anne and Blair, Lynne and Seward, Derek Accurate estimation of forward path geometry using two-clothoid road model /entry/357 357 Wed, 31 Dec 1969 16:33:31 -0800 Khosla, D The 1-Hyperbolic Projection for User Interfaces /entry/358 The problem of dealing with representations of information that does not fit conveniently within allotted screen space is pervasive in graphical interfaces. While there are techniques for dealing with this problem in various ways, some properties of such existing techniques are not satisfying. For example, global structure of information may be lost in favor of local focus, or information may not be mapped into a rectangular area. The 1-hyperbolic interface is proposed to deal with some of these deficiencies, and the mathematics involved in display and interaction are derived. The calculations necessary for this interface are easy to implement, and can run reasonably even on slow devices. A fully functional prototype for displaying tree structures has been developed to compare the effects of this new interface to those of a standard interface. The results of usability experiments conducted with this prototype are also presented and analyzed. 358 Wed, 31 Dec 1969 16:33:31 -0800 Kolliopoulos, Alexander Drawing and Labeling High-Quality Metro Maps by Mixed-Integer Programming. /entry/359 Metro maps are schematic diagrams of public transport networks that serve as visual aids for route planning and navigation tasks. It is a challenging problem in network visualization to automatically draw appealing metro maps. There are two aspects to this problem that depend on each other: the layout problem of finding station and link coordinates and the labeling problem of placing non-overlapping station labels. In this paper we present a new integral approach that solves the combined layout and labeling problem (each of which, independently, is known to be NP-hard) using mixed-integer programming (MIP). We identify seven design rules used in most real-world metro maps. We split these rules into hard and soft constraints and translate them into a MIP model. Our MIP formulation finds a metro map that satisfies all hard constraints (if such a drawing exists) and minimizes a weighted sum of costs that correspond to the soft constraints. We have implemented the MIP model and present a case study and the results of an expert assessment to evaluate the performance of our approach in comparison to both manually designed official maps and results of previous layout methods. 359 Wed, 31 Dec 1969 16:33:31 -0800 Nöllenburg, Martin and Wolff, Alexander A Paradoxical Property of the Monkey Book /entry/360 A "monkey book" is a book consisting of a random distribution of letters and blanks, where a group of letters surrounded by two blanks is defined as a word. We compare the statistics of the word distribution for a monkey book with the corresponding distribution for the general class of random books, where the latter are books for which the words are randomly distributed. It is shown that the word distribution statistics for the monkey book is different and quite distinct from a typical sampled book or real book. In particular the monkey book obeys Heaps' power law to an extraordinary good approximation, in contrast to the word distributions for sampled and real books, which deviate from Heaps' law in a characteristics way. The somewhat counter-intuitive conclusion is that a "monkey book" obeys Heaps' power law precisely because its word-frequency distribution is not a smooth power law, contrary to the expectation based on simple mathematical arguments that if one is a power law, so is the other. 360 Wed, 31 Dec 1969 16:33:31 -0800 Bernhardsson, Sebastian and Baek, Seung Ki and Minnhagen, Petter A Note on Approximating the Normal Distribution Function /entry/361 361 Wed, 31 Dec 1969 16:33:31 -0800 Aludaat, K M and Alodat, M T An example of a computable absolutely normal number /entry/362 362 Wed, 31 Dec 1969 16:33:31 -0800 Figueira, Santiago Theoretical Computer Science Cheat Sheet /entry/363 363 Wed, 31 Dec 1969 16:33:31 -0800 Steve Seiden Tree automata techniques and applications /entry/364 364 Wed, 31 Dec 1969 16:33:31 -0800 Comon, Hubert and Dauchet, M and Gilleron, R Automatic calculation of plane loci using Grobner bases and integration into a Dynamic Geometry System /entry/365 365 Wed, 31 Dec 1969 16:33:31 -0800 Gerh, Michael Juggling Probabilities /entry/366 366 Wed, 31 Dec 1969 16:33:31 -0800 Warrington, Gregory S. The isoperimetric problem /entry/367 367 Wed, 31 Dec 1969 16:33:31 -0800 Blasjo, Viktor Hierarchical Position Based Dynamics /entry/368 368 Wed, 31 Dec 1969 16:33:31 -0800 Faure, F. and Teschner, M. James Garfield's Proof of the Pythagorean Theorem /entry/369 369 Wed, 31 Dec 1969 16:33:31 -0800 Ellermeyer, S F Surreal Numbers – An Introduction /entry/370 370 Wed, 31 Dec 1969 16:33:31 -0800 Tøndering, Claus A discursive grammar for customizing mass housing: the case of Siza's houses at Malagueira /entry/371 371 Wed, 31 Dec 1969 16:33:31 -0800 Duarte, J Hypercomputation: computing more than the Turing machine /entry/372 Due to common misconceptions about the Church-Turing thesis, it has been widely assumed that the Turing machine provides an upper bound on what is computable. This is not so. The new field of hypercomputation studies models of computation that can compute more than the Turing machine and addresses their implications. In this report, I survey much of the work that has been done on hypercomputation, explaining how such non-classical models fit into the classical theory of computation and comparing their relative powers. I also examine the physical requirements for such machines to be constructible and the kinds of hypercomputation that may be possible within the universe. Finally, I show how the possibility of hypercomputation weakens the impact of Godel's Incompleteness Theorem and Chaitin's discovery of 'randomness' within arithmetic. 372 Wed, 31 Dec 1969 16:33:31 -0800 Ord, Toby Comparison of geometric figures /entry/373 373 Wed, 31 Dec 1969 16:33:31 -0800 Glenis, Spyros and Kapovich, M. and Brodskiy, N. and Dydak, J. and Lang, U. and Ballinger, B. and Blekherman, G. and Cohn, H. and Giansiracusa, N. and Kelly, E. and Others On a curious property of 3435 /entry/374 Folklore tells us that there are no uninteresting natural numbers. But some natural numbers are more interesting then others. In this article we will explain why 3435 is one of the more interesting natural numbers around. We will show that 3435 is a Munchausen number in base 10, and we will explain what we mean by that. We will further show that for every base there are finitely many Munchausen numbers in that base. 374 Wed, 31 Dec 1969 16:33:31 -0800 Berkel, Daan Van Better approximations to cumulative normal functions /entry/375 375 Wed, 31 Dec 1969 16:33:31 -0800 West, Graeme Continued fraction algorithms, functional operators, and structure constants /entry/376 376 Wed, 31 Dec 1969 16:33:31 -0800 Flajolet, P. and Vallée, B. Can One Hear the Shape of a Drum? /entry/377 377 Wed, 31 Dec 1969 16:33:31 -0800 Kac, Mark A history of mathematical notations /entry/378 378 Wed, 31 Dec 1969 16:33:31 -0800 Florian Cajori Interpolating Solid Orientations with a $C^2$ -Continuous B-Spline Quaternion Curve /entry/379 379 Wed, 31 Dec 1969 16:33:31 -0800 Ge, Wenbing and Huang, Zhangjin and Wang, Guoping A classification for shaggy dog stories /entry/380 380 Wed, 31 Dec 1969 16:33:31 -0800 Brunvand, J.H. Animating rotation with quaternion curves /entry/381 Solid bodies roll and tumble through space. In computer animation, so do cameras. The rotations of these objects are best described using a four coordinate system, quaternions, as is shown in this paper. Of all quaternions, those on the unit sphere are most suitable for animation, but the question of how to construct curves on spheres has not been much explored. This paper gives one answer by presenting a new kind of spline curve, created on a sphere, suitable for smoothly in-betweening (i.e. interpolating) sequences of arbitrary rotations. Both theory and experiment show that the motion generated is smooth and natural, without quirks found in earlier methods. 381 Wed, 31 Dec 1969 16:33:31 -0800 Shoemake, Ken Spontaneous knotting of an agitated string. /entry/382 It is well known that a jostled string tends to become knotted; yet the factors governing the "spontaneous" formation of various knots are unclear. We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots. Above a critical string length, the probability P of knotting at first increased sharply with length but then saturated below 100%. This behavior differs from that of mathematical self-avoiding random walks, where P has been proven to approach 100%. Finite agitation time and jamming of the string due to its stiffness result in lower probability, but P approaches 100% with long, flexible strings. We analyzed the knots by calculating their Jones polynomials via computer analysis of digital photos of the string. Remarkably, almost all were identified as prime knots: 120 different types, having minimum crossing numbers up to 11, were observed in 3,415 trials. All prime knots with up to seven crossings were observed. The relative probability of forming a knot decreased exponentially with minimum crossing number and Möbius energy, mathematical measures of knot complexity. Based on the observation that long, stiff strings tend to form a coiled structure when confined, we propose a simple model to describe the knot formation based on random "braid moves" of the string end. Our model can qualitatively account for the observed distribution of knots and dependence on agitation time and string length. 382 Wed, 31 Dec 1969 16:33:31 -0800 Raymer, Dorian M and Smith, Douglas E Four questions about fuzzy rankings /entry/383 383 Wed, 31 Dec 1969 16:33:31 -0800 Brian Hayes On Buffon Machines and Numbers /entry/384 The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically "computes" the number 2/pi ~ 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithmic-series distributions. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities of success are expressible in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5). Generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to design Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function. 384 Wed, 31 Dec 1969 16:33:31 -0800 Flajolet, Philippe Zaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle /entry/385 385 Wed, 31 Dec 1969 16:33:30 -0800 Andrew Granville Implications of the Turing Completeness of Reaction-Diffusion Models, informed by GPGPU simulations on an XBox 360: Cardiac Arrythmias, Re-entry and the Halting Problem /entry/386 386 Wed, 31 Dec 1969 16:33:30 -0800 Scarle, S openttd logic gates /entry/387 Here's a rather old (and probably outdated) look at how one could simulate digital logic circuits with OpenTTD. Includes the fastest four-bit ripple-carry adder ever: takes about two months (of in-game time) for the carry information to propagate. 387 Wed, 31 Dec 1969 16:33:30 -0800 Heikki Kallasjoki Push-pull LEGO logic gates /entry/388 388 Wed, 31 Dec 1969 16:33:30 -0800 Randomwraith Misconceptions about the Golden Ratio /entry/389 389 Wed, 31 Dec 1969 16:33:30 -0800 Markowsky, George On Furstenberg's Proof of the Infinitude of Primes /entry/390 390 Wed, 31 Dec 1969 16:33:30 -0800 Mercer, Idris D What symmetry groups are present in the Alhambra? /entry/391 391 Wed, 31 Dec 1969 16:33:30 -0800 Grünbaum, Branko The Origin of Chemical Elements /entry/392 392 Wed, 31 Dec 1969 16:33:30 -0800 Alpher, R. and Bethe, H. and Gamow, G. The role of instrumental and relational understanding in proofs about group isomorphisms /entry/393 393 Wed, 31 Dec 1969 16:33:30 -0800 Weber, K. A Note on Boolos' Proof of the Incompleteness Theorem /entry/394 We give a proof of Gödel's first incompleteness theorem based on Berry's paradox, and from it we also derive the second incompleteness theorem model-theoretically. 394 Wed, 31 Dec 1969 16:33:30 -0800 Kikuchi, Makoto A Closed-Form Algorithm for Converting Hilbert Space-Filling Curve Indices /entry/395 395 Wed, 31 Dec 1969 16:33:30 -0800 Chen, Chih-sheng and Lin, Shen-yi and Fan, Min-hsuan and Huang, Chua-huang Hilbert R-tree: An improved R-tree using fractals /entry/396 396 Wed, 31 Dec 1969 16:33:30 -0800 Kamel, Ibrahim and Faloutsos, Christos Digital halftoning space filling curves /entry/397 397 Wed, 31 Dec 1969 16:33:30 -0800 Luiz C. Velho and Jonas M. Gomes A game for budding knot theorists /entry/398 398 Wed, 31 Dec 1969 16:33:30 -0800 Dave Richeson On Mathematics and Mathematicians /entry/399 ON MATHEMATICS AND MATHEMATICIANS Formerly titled Memorabilia Mathematica or the Philomathss Quotation-Book By Robert Edouard Moritz DOVER PUBLICATIONS INC., NEW YORK Copyright 1914 by Robert Edouard Moritz Copyright 1942 by Cassia K. Moritz This new Dover edition first published in 1958 is an unabridged and unaltered republication of the first edition which was originally en titled Memorabilia, Mathematical, or The Philo maths Quotation-Book. Manufactured in the United States of America Dover Publications, Inc. 920 Broadway New York 10, N. Y. PREFACE EVERY one knows that the fine phrase God geometrizes is attributed to Plato, but few know where this famous passage is found, or the exact words in which it was first expressed. Those who, like the author, have spent hours and even days in the search of the exact statements, or the exact references, of similar famous passages, will not question the timeliness and usefulness of a book whose distinct purpose it is to bring together into a single volume exact quotations, with their exact references, bearing on one of the most time-honored, and even today the most active and most fruitful of all the sciences, the queen mother of all the sciences, that is, mathematics. It is hoped that the present volume will prove indispensable to every teacher of mathematics, to every writer on mathe matics, and that the student of mathematics and the related sciences will find its perusal not only a source of pleasure but of encouragement and inspiration as well. The layman will find it a repository of useful information covering a field of knowledge which, owing to the unfamiliar and hence repellant character of the language employed by mathematicians, ispeculiarly in accessible to the general reader. No technical processes or technical facility is required to understand and appreciate the wealth of ideas here set forth in the words of the worlds great thinkers. No labor has been spared to make the present volume worthy of a place among collections of a like kind in other fields. Ten years have been devoted to its preparation, years, which if they could have been more profitably, could scarcely have been more pleasurably employed. As a result there have been brought together over one thousand more or less familiar passages pertaining to mathematics, by poets, philosophers, historians, statesmen, scientists, and mathematicians. These have been gathered from over three hundred authors, and have been vi PREFACE grouped under twenty heads, and cross indexed under nearly seven hundred topics. The authors original plan was to give foreign quotations both in the original and in translation, but with the growth of mate rial this plan was abandoned as infeasible. It was thought to serve the best interest of the greater number of English readers to give translations only, while preserving the references to the original sources, so that the student or critical reader may readily consult the original of any given extract. In cases where the translation is borrowed the translators name is inserted in brackets immediately after the authors name. Brackets are also used to indicate inserted words or phrases made necessary to bring out the context. The absence of similar English works has made the authors work largely that of the pioneer. Rebi res Math6matiques et Math naticiens and Ahrens Scherz und Ernst in der Mathematik have indeed been frequentlyconsulted but rather with a view to avoid overlapping than to receive aid. Thus certain topics as the correspondence of German and French mathematicians, so excellently treated by Ahrens, have pur posely been omitted. The repetitions are limited to a small number of famous utterances whose absence from a work of this kind could scarcely be defended on any grounds. No one can be more keenly aware of the shortcomings of a work than its author, for none can have so intimate an acquaint ance with it... 399 Wed, 31 Dec 1969 16:33:30 -0800 Moritz, Robert Edoward ARTIFICIAL NEURAL NETWORK MODELING OF APPLE DRYING PROCESS /entry/400 400 Wed, 31 Dec 1969 16:33:30 -0800 KHOSHHAL, ABBAS and DAKHEL, ASGHAR ALIZADEH and ETEMADI, AHMAD and ZERESHKI, SINA Detection of transposition errors in decimal numbers /entry/401 401 Wed, 31 Dec 1969 16:33:30 -0800 Freeman, H Circle Packing for Origami Design Is Hard /entry/402 We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems. On the positive side, we show that any set of circles of total area 1 can be packed into a square of size 8/pi=2.546... These results are motivated by problems arising in the context of origami design. 402 Wed, 31 Dec 1969 16:33:30 -0800 Demaine, E.D. and Fekete, S.P. and Lang, R.J. What Sequential Games , the Tychonoff Theorem and the Double-Negation Shift have in Common /entry/403 This is a tutorial for mathematically inclined functional programmers, based on previously published, peered reviewed theoretical work. We discuss a higher-type functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a computational version of the Tychonoff Theorem from topology, and (3) realizes the Double Negation Shift from logic and proof theory. The functional makes sense for finite and infinite (lazy) lists, and in the binary case it amounts to an operation that is available in any (strong) monad. In fact, once we define this monad in Haskell, it turns out that this amazingly versatile functional is already available in Haskell, in the standard prelude, called sequence, which iterates this binary operation. Therefore Haskell proves that this functional is even more versatile than anticipated, as the function sequence was introduced for other purposes by the language designers, in particular the iteration of a list of monadic effects (but effects are not what we discuss here) 403 Wed, 31 Dec 1969 16:33:30 -0800 Oliva, Paulo and Escardo, Martin The Mathematics of Musical Instruments /entry/404 404 Wed, 31 Dec 1969 16:33:30 -0800 Hall, Rachel W. and Josic, Kresimir The liouville-heath-brown-zagier proof of the two squares theorem and generalizations /entry/405 Heath-Brown [6] suggested a short proof of the two squares theorem, thereby simplifying ideas of Liouville. Zagier [15] suggested a particularly neat form of this, a "One sentence proof". It consists of two suitable involutions on the finite set of the solutions of p = x 2 +4yz in positive integers. A parity argument ensures the existence of a solution with y = z. The proof can be stated in one sentence since elementary calculations (namely to check that the mappings are well defined and are indeed involutory) can be left to the reader. The proof remained somewhat mysterious, since it is not obvious, where these mappings come from. In this paper we reveal this mystery and systematically explore similar proofs that can be given for related problems. We show that the very same method proves results on p = x 2 +2y 2 (see also Jackson [7]), p = x 2 2y 2 , p = 3x 2 + 4y 2 , and p = 3x 2 4y 2 . 405 Wed, 31 Dec 1969 16:33:30 -0800 Elsholtz, Christian Constructive gem: juggling exponentials /entry/406 406 Wed, 31 Dec 1969 16:33:30 -0800 Bauer, Andrej ‘Knowable' As ‘Known After an Announcement' /entry/407 407 Wed, 31 Dec 1969 16:33:30 -0800 Balbiani, Philippe and Baltag, Alexandru and Ditmarsch, Hans Van and Herzig, Andreas and Hoshi, Tomohiro and De Lima, Tiago Foolproof : A Sampling of Mathematical Folk Humor /entry/408 408 Wed, 31 Dec 1969 16:33:30 -0800 Renteln, Paul and Dundes, Alan A formal system for Euclid's Elements /entry/409 We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning. 409 Wed, 31 Dec 1969 16:33:30 -0800 Avigad, Jeremy and Dean, Edward and Mumma, John An aperiodic hexagonal tile /entry/410 We show that a single tile can fill space uniformly but not admit a periodic tiling. The space--filling tiling that can be built from copies of the tile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. We present the tile, prove that the tilings it admits are not periodic, and discuss some of their remarkable properties, including their relation to a previously known tiling. We also clarify some subtleties in the definitions of the terms "nonperiodic tiling" and "aperiodic tile". For a reasonable interpretation of these terms, the tile presented here is the only known example of an aperiodic tile. 410 Wed, 31 Dec 1969 16:33:30 -0800 Socolar, Joshua E. S. and Taylor, Joan M. How to explain zero-knowledge protocols to your children /entry/411 411 Wed, 31 Dec 1969 16:33:30 -0800 Quisquater, JJ and Quisquater, M Plane recursive trees, Stirling permutations and an urn model /entry/412 412 Wed, 31 Dec 1969 16:33:30 -0800 Janson, Svante Unbounded spigot algorithms for the digits of pi /entry/413 413 Wed, 31 Dec 1969 16:33:30 -0800 Gibbons, J.