The Rotating Table
Ted G. Lewis, Stephen Willard · Mathematics Magazine · 1980
Imagine a square table at each corner of which is a deep well. Hidden from view at the bottom of each well is a drinking glass which may be either upright or inverted. You player) may place one or both hands into any wells desired and may adjust the glasses found there in any way you wish. After you have done this, if all glasses around the table are in the same state, a rings and you win. Otherwise, the table is rotated (with you blindfolded) and you are allowed again to select wells and adjust glasses. The question: can you force the to ring in a finite number of moves? This problem can be generalized in an obvious way: Assume a polygonal table with n wells and a player (the bell ringing octupus) with no<n hands. We will show here that, in this general setting, a player can force the objective of ringing the in a finite number of moves if and only if he has been provided with at least [(p 1)/p]n hands, where p is the largest prime divisor of n. The problem for the table with 4 wells has been discussed recently by Gardner ([2], [3]). In [3] he reports that R. L. Graham and P. Diaconis have shown that the n-well game can be won by a player with n -2 hands if and only if n is nonprime. It is also suggested that, for composite n, fewer than n -2 hands may be required. That this is indeed the case follows from the result we have stated above. In what follows, we will establish the necessity of [(p 1)/p]n hands for the n-well table, and then the sufficiency of [(p 1)/p]n hands. We will also remark on the number of moves required to win the game (expanded to encompass a discussion of the total energy required to win), and on a further generalization of the game to one in which the glasses are replaced by objects, each of which can assume any of h different states. It will be convenient to think of the n-well game as being played on a regular n-gon (vertex=well) between two contestants: the player, who has been provided with no<n hands, and fate, who initially sets the glasses in the wells and who at all times knows the state of every glass. A move of the game proceeds as follows: The player places his hands over the wells