This variant is almost similar to the normal nim game which states -
Two players take turns to remove one or more items from a single, non-empty pile. The player who removes the last item from the last non-empty pile is declared as a winner.
The twist is an inclusion of a "pass" move for each non-empty pile. To make myself clear, a player can choose to pass his move on any non-empty pile on which there was no "pass" move made.
I looked up grundy numbers to solve this problem, but I do not how to add the rule of pass move.
$G(0) = mex(empty-set) = 0$
$G(1) = mex(G(1),0) = ?$
mex(g) is a function which takes a set of non-negative integers and outputs smallest non-negative integer that is not present in the list.
For ex :
mex(0,1,2) = 3 and
mex(0,2,3) = 1
I just don't know how to proceed. I'd appreciate some help. Also, I'd like to add that the number of piles is small say 10-15 but the number contained in each pile is quite huge. Hence those recursive backtracking type of solutions won't work.
Update : The pass move is not mandatory.