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A Mathematical Analysis of The Generalized Oval Track Puzzle

The oval track puzzle (also known as Top Spin) is a game consisting of 20 numbered tiles in an oval shaped track. Also, there is a fixed window (the swapping window) of 4 tiles that reverses the order of the tiles within the window, leaving the other 16 tiles fixed. The object of the puzzle is to reorder the tiles into counting order using the mechanisms of the puzzle. Previously, conditions for both solvability and non-solvability for the generalized oval track puzzle with n total tiles and k tiles in the swapping window were shown. We will now prove tight asymptotic bounds on the number of swaps needed to solve any configuration of a puzzle with n total tiles and k tiles in the swapping window provided that n and k yield a solvable case to begin with. These bounds will be asymptotic because we will assume that n grows infinitely and k stays fixed. RHIT Undergrad. Math. J., Vol. 12, no. 1 Page 71