5-pile Same-Nim

The game Same-Nim is a two player game played with piles of stones. A valid move consist of removing a positive number of stones from one of the piles, such that there is some pair of piles that contains the same number of stones. The players alternate turns to make a move. A player loses the game if they are unable to remove any stones on their turn.

A game of Same-Nim is played with 5 piles of sizes \((10,10,a,b,c)\), where \(a, b, c \) are integers from 1 to 10 inclusive. For how many of the \(1000\) possible ordered triples \((a,b,c)\) does the first player have a winning strategy?

Details and assumptions

A valid move may result in more than 1 pair of piles that contain the same number of stones.

The pair of piles with the same number of stones may both have \(0\) stones, if the piles had stones in them when the game started.


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