A problem by Sreejato Bhattacharya

Level pending

Calvin and Lino play a modified game of Nim. First, they randomly choose a positive integer \(n\) between \(10\) and \(100\) (inclusive). They then take a pile of stones having \(n\) stones. The players take turns alternatively removing a certain number of stones from the pile. The rules are:

  • At the first move, the first player removes a positive integer number of stones, but he cannot remove \(10\) stones or more.

  • At the consequent moves, each player has to remove a positive integer number of stones that is at most \(10\) times the number of stones removed by the other player in the last turn.

  • The player who removes the last stone wins.

Calvin goes first. Assuming each player plays optimally, the probability that Lino wins the game can be expressed as \(\dfrac{a}{b}\), where \(a, b\) are positive coprime integers. Find \(a+b\).

Details and assumptions

At each move, a player has to remove a positive number of stones. Removing nothing from the pile isn't considered as a valid move.


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