# How do you fix an Oval Track Puzzle?

• Published in 2016
In the collections
The oval track group, $OT_{n,k}$, is the subgroup of the symmetric group, $S_n$, generated by the basic moves available in a generalized oval track puzzle with $n$ tiles and a turntable of size $k$. In this paper we completely describe the oval track group for all possible $n$ and $k$ and use this information to answer the following question: If the tiles are removed from an oval track puzzle, how must they be returned in order to ensure that the puzzle is still solvable? As part of this discussion we introduce the parity subgroup of $S_n$ in the case when $n$ is even.

### BibTeX entry

@article{HowdoyoufixanOvalTrackPuzzle,
title = {How do you fix an Oval Track Puzzle?},
abstract = {The oval track group, {\$}OT{\_}{\{}n,k{\}}{\$}, is the subgroup of the symmetric group,
{\$}S{\_}n{\$}, generated by the basic moves available in a generalized oval track
puzzle with {\$}n{\$} tiles and a turntable of size {\$}k{\$}. In this paper we completely
describe the oval track group for all possible {\$}n{\$} and {\$}k{\$} and use this
information to answer the following question: If the tiles are removed from an
oval track puzzle, how must they be returned in order to ensure that the puzzle
is still solvable? As part of this discussion we introduce the parity subgroup
of {\$}S{\_}n{\$} in the case when {\$}n{\$} is even.},
url = {http://arxiv.org/abs/1612.04476v3 http://arxiv.org/pdf/1612.04476v3},
year = 2016,
author = {David A. Nash and Sara Randall},
comment = {},
urldate = {2018-03-13},
archivePrefix = {arXiv},
eprint = {1612.04476},
primaryClass = {math.GR},
collections = {Easily explained,Puzzles}
}