How do you fix an Oval Track Puzzle?

TL;DR

This paper characterizes the algebraic structure of the oval track puzzle group for all configurations and determines how to restore tiles to keep the puzzle solvable, introducing a new parity subgroup for even cases.

Contribution

It provides a complete description of the oval track group for all n and k, and addresses tile restoration for solvability, including a new parity subgroup for even n.

Findings

Complete group descriptions for all n and k

Conditions for tile restoration to maintain solvability

Introduction of the parity subgroup for even n

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.