# Modification of Nim Game Winning Strategy

Here is the variation: We have some P piles of numbers,each having some pi numbers , and in each turn a player may choose a pile, and then a number from the chosen pile. All the numbers greater than equal to the chosen number are removed from the chosen pile. We have to determine, if the player moving first has a winning strategy.

What I have done is, read about the NIM game and how we could generate a winning strategy. But this variation seems to have a dynamic rule set. Is this a standard type? If so, where can I read about this.? If not, how to approach this? Help in any direction will be appreciated.

• I think that it is equivalent to a standard NIM game with P piles of objects (not numbers). Indeed without loss of generality, we can suppose that the numbers are ordered. Then it is easy to see that a pile of numbers with some "holes" (e.g. 1,2,7,8} is equivalent to a pile in which the numbers are compacted (e.g. 1,2,3,4). But a pile with $n$ ordered numbers without holes, in which we pick a number and delete all numbers greater to it is equivalent to a pile of $n$ objects in which we decide how many objects must be removed. – Vor Nov 3 '13 at 11:12
• @Vor I had similar thoughts, like if a pile is {1,2,3,4}, this could be converted to a pile of objects, and the allowed removals are {4,3,2,1} objects respectively. Is that what you're coming at? – Salena Nov 3 '13 at 12:26
• yes, it is equivalent to a standard multi-heap nim – Vor Nov 3 '13 at 12:29
• It seems to be very hard to determine a winning strategy because you haven't told us what constitutes "winning". – gnasher729 Jul 4 '17 at 10:40