I have the following question with me:
"Start with several piles of chips. Two players move alternately. A move consists in splitting every pile with more than one chip in two piles. The one who makes the last move wins. For what initial conditions does the first player win and what is his winning strategy"
I found this similar to Grundy's game but they haven't mentioned in the given question that the two piles are unequal. My book says that the first player wins if the largest number of chips in a pile is not equal to $2^m - 1$ for some $m$. The book further says that the required strategy for the first player is to split the piles in such a way that the one of the resulting piles should have the number of chips equal to $2^m - 1$ for some.
However, let's say I have $(6,2,1)$ to start with.
I perform the following operations $$(6,2,1) \rightarrow (3,3,2,1)$$This move is made by the first player. It is quite easy to see that B wins in any case in the above situation. Thus, followig the given strategy the statement given in the book seems to be wrong. Is there any mistake with the book or is there any mistake with my interpretation? A solution with the correct conditions would also be appreciated.
Thanks in advance