We know how to win the classic regular Nim (two players) Classic rules: Any number of beans into any number of separate piles Each move, the player whose turn it is, must choose one pile of beans and remove anywhere from one bean to all the beans in only ONE pile Player taking the last bean looses (as example).
I am looking for the correct strategy for the following variation: Same rules as in the classic regular Nim EXCEPT THAT only once in the game, a player, and only one, MAY PASS his turn.
I found some winning positions 2 1,1 1,N,N (with N>1) 2,3,5 4,5,8, 6,7,3
and I believe that, before the Pass is used, we need to use one "virtual" pile with one bean but I could not find a general strategy.