# 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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