Abstract:

We study a variation of the combinatorial game of 2-pile Nim. Move as in 2-pile Nim but with the following constraint:

Provided the previous player has just removed say $x>0$ tokens from the pile with a less number of tokens, the next player may remove $x$ tokens from the pile with more tokens. But for each move, in ``a strict sequence of previous player - next player moves'', such an imitation takes place, the value of an imitation counter is increased by one unit. As this counter reaches a predetermined natural number, then by the rules of this game, if the previous player once again removes a positive number of tokens from the pile with less tokens, the next player may not remove this same number of tokens from the pile with more tokens.

We show that the winning positions of this game in a sense resembles closely the winning positions of the game of Wythoff Nim - more precisely a version of Wythoff Nim with a Muller twist. In fact, we show a slightly more general result in which we have relaxed the notion of what an imitation is.

As a round off, given that the first player knows the winning strategy for a particular game, we discuss a successful "learning" strategy for the second player.