Rotating-Table Games and Derivatives of Words

We consider two versions of a game for two players, A and B. The game consists of manipulations of words of length n over an alphabet of size (j, for arbitrary n and (j. For (j = 2 the game is described as follows: Initially, player A puts n drinking glasses on a round table, some of which are upside down. Player B attempts to force player A to set all the glasses in the upright position. For this, he instructs player A to invert some of the glasses. Before following the instruction, player A has the freedom to rotate the table, and then to inverts the glasses that are in the locations originally pointed by player B. In one version of the game player B is blindfolded, and in the other he is not. We show that player B has winning strategies for both games iff n and (j are powers of the same prime. In boths games we provide optimal winning strategies for B. The analysis of the games is closely related to the concept of the derivative of a (j-ary word oflength n. In particular, it is related to the depth of such word, which is the smallest k such that the k-th derivative of the word is the all-zero word. We give tight upper bounds on the depth of (j-ary words of length n, where (j and n are powers of the same prime.