Winning strategy for add game

Question

I was thinking in this kind of game:

Two players start from 0 and alternately add a number from 1 to 10 to the sum. The player who reaches 100 wins. The winning strategy is to reach a number in which the digits are subsequent (e.g., 01, 12, 23, 34,...) and control the game by jumping through all the numbers of this sequence. Once a player reaches 89, the opponent can only choose numbers from 90 to 99, and the next answer can in any case be 100.

And I wonder if there is a winning strategy for a number $n$ and any maximum number that can be added

Answer 1

Let the final number be $n$ and the current number be $i \leq n$. Then the player whose turn it is will win with perfect play iff $n - i$ is a multiple of 11.

Proof: Suppose $n - i = k \cdot 11$. It is clear that the non-current player will always win by picking $11 - z$ immediately after the current player picks $z$.

Now suppose $n - i$ is not a multiple of 11. That is, pick $k$ such that $n - i \equiv k \mod 11$ and $1 \leq k \leq 10$. Then the current player should play $k$; by the previous direction, the current player is then guaranteed a victory.

So if we start at $0$ and end at $n$, player 2 wins iff $n$ is a multiple of 11.