4-pile Same-Nim

Probability Level 5

The game Same-Nim is a two player game played with piles of stones. A valid move consists 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 4 piles of sizes (30,30,a,b),(30,30,a,b), where 1a,b30.1 \leq a, b \leq 30. For how many of the 900900 possible ordered pairs (a,b)(a,b) 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 00 stones, if the piles had stones in them when the game started.


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