# Let he who has no sin cast the first stone

Calvin and Lino are playing a game with 4 piles of stones. At the start of the game, Calvin rolls four 8-sided dice, each with the numbers 0 through 7 on them, to help in determining the sizes of the piles of stones. Pile $$i$$ will have size $$10^i + d_i,$$ where $$d_i$$ is the number rolled on the $$i$$th die.

Players take turns removing stones from the piles, and Lino gets to make the first move. On a player's turn, that player is allowed to remove a number of stones that is a power of 2 from one of the piles. The last player who makes a move is the winner.

Assuming that Calvin and Lino both play optimally, the probability that Lino is able to remove 4 stones from a pile on his first move and still win the game can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b$$?

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