The Puzzle No. Crunch (TopSpin)and Word Length in the Symmetric Group

Introduction

The puzzle No. Crunch was made by Binary Arts, which company changed its name to ThinkFun and produced the puzzle under the name TopSpin (but the puzzle is no longer listed at ThinkFun'sWeb site [2009]). The puzzle (Figure 1) contains 20 discs numbered 1 through 20, which sit on an oval track and can slide in either direction. At the top of the track is a turnstile that when turned 180◦, as demonstrated in Figure 1, will reverse the order of four adjacent discs. The object of the puzzle is to arrange the numbered discs into some specific order by sliding and using the turnstile.

The puzzle was invented by Ferdinand Lammertink and patented in the U.S. in 1989 [U.S. Patent and Trademark Office 1989]. It can be played online at various sites (Scherphuis [n.d.], van Leeuwen [n.d.], Vermaak [n.d.], Toshio [n.d.]), with a widget version for Nokia phones [Silvennoinen 2008]. A video of a proficient solver solving an instance of the puzzle (under another brand name) in under two minutes is at available online [Top Spin. . . 2008]. Scherphuis mentions a larger 1997 puzzle from Taiwan entitled Educational LogicGame, which has 26 pieces in the loop and two small sidetracks whose use is optional [U.S. Patent and Trademark Office 1997]. This latter puzzle (without the sidetracks) is available as a free computer game (forWindows) under the name Top Spin [Archer 2005].
Both Bennett [2000] and Kiltinen [2003] studied the original puzzle extensively using permutations. Each showed that all possible disc arrangements are obtainableand presented a sequence ofmoves to solve thepuzzle. Kiltinen's book also comes with a CD that simulates the puzzle.
The puzzle has been a testbed for research in artificial intelligence, particularly in heuristics for pattern databases [Felner 2005; Felner et al. 2005; Yang et al. 2008].