# The no-three-in-line problem on a torus

- Published in 2012
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Let $T(\mathbb{Z}_m \times \mathbb{Z}_n)$ denote the maximal number of points that can be placed on an $m \times n$ discrete torus with "no three in a line," meaning no three in a coset of a cyclic subgroup of $\mathbb{Z}_m \times \mathbb{Z}_n$. By proving upper bounds and providing explicit constructions, for distinct primes $p$ and $q$, we show that $T(\mathbb{Z}_p \times \mathbb{Z}_{p^2}) = 2p$ and $T(\mathbb{Z}_p \times \mathbb{Z}_{pq}) = p+1$. Via Grobner bases, we compute $T(\mathbb{Z}_m \times \mathbb{Z}_n)$ for $2 \leq m \leq 7$ and $2 \leq n \leq 19$.

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### BibTeX entry

@article{Thenothreeinlineproblemonatorus, title = {The no-three-in-line problem on a torus}, author = {Jim Fowler and Andrew Groot and Deven Pandya and Bart Snapp}, url = {http://arxiv.org/abs/1203.6604v1 http://arxiv.org/pdf/1203.6604v1}, urldate = {2019-12-10}, year = 2012, abstract = {Let {\$}T(\mathbb{\{}Z{\}}{\_}m \times \mathbb{\{}Z{\}}{\_}n){\$} denote the maximal number of points that can be placed on an {\$}m \times n{\$} discrete torus with "no three in a line," meaning no three in a coset of a cyclic subgroup of {\$}\mathbb{\{}Z{\}}{\_}m \times \mathbb{\{}Z{\}}{\_}n{\$}. By proving upper bounds and providing explicit constructions, for distinct primes {\$}p{\$} and {\$}q{\$}, we show that {\$}T(\mathbb{\{}Z{\}}{\_}p \times \mathbb{\{}Z{\}}{\_}{\{}p^2{\}}) = 2p{\$} and {\$}T(\mathbb{\{}Z{\}}{\_}p \times \mathbb{\{}Z{\}}{\_}{\{}pq{\}}) = p+1{\$}. Via Grobner bases, we compute {\$}T(\mathbb{\{}Z{\}}{\_}m \times \mathbb{\{}Z{\}}{\_}n){\$} for {\$}2 \leq m \leq 7{\$} and {\$}2 \leq n \leq 19{\$}.}, comment = {}, archivePrefix = {arXiv}, eprint = {1203.6604}, primaryClass = {math.CO}, collections = {Fun maths facts,Geometry,Puzzles} }