6. Cube puzzles
The goal of the puzzle is to move all of the rings from the initial pole to one of the other poles so that no ring of larger diameter is ever placed on one of smaller diameter and one ring at a time is moved. Solving the problem with 3 rings is not difficult (it involves 7 moves); the 4-ring version of the Tower of Hanoi requires 15 moves to solve. This might suggest a possible connection with the n-cube, because the numbers are one less than the number of vertices in ann-cube where n = 3 and n = 4!
For a variety of reasons HC's on an n-cube are of interest. A natural question is to ask how many different HC's there are on then-cube. The crux of the issue, as is typically the case in enumeration problems, is the meaning of the term "different." The usual definition of different for Hamiltonian circuits is that they use different edges. However, for a very symmetric object like the cube, one might want to consider two HC's different if no symmetry of the cube transfers one HC into another.
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