A problem by Sreejato Bhattacharya

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.