This game starts with a positive integer. Two players take turns to remove a single digit from that integer. After the digit is removed any resulting leading zeros are removed.
For example, removing a digit from $105$ results in either $5$, $10$ or $15$.
The winner is the person who removes the last nonzero digit.
Define $W(N)$ to be how many positive integers less than $N$ for which the first player can guarantee a win given optimal play. You are given $W(100) = 18$ and $W(10^4) = 1656$.
Find $W(10^{18})$.
Solution
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