# Two players choosing one of three numbers

$$A$$ and $$B$$ play the following game. Initially, for positive integer $$n$$, each player takes turns choosing one of three numbers:

• $$1$$

• the number of digits of $$n$$

• the sum of the digits of $$n$$.

and then updates the number $$n$$ by subtracting the number selected. If $$n$$ becomes $$0$$, the person who has that turn wins. Who will be the winner if $$A$$ is always the first to go?

For example:

• With $$n=3$$, the result is $$A$$, because $$A$$ chooses the number that is the sum of the digits of $$n = 3$$, after selecting, $$n$$ is updated and becomes $$0$$. So $$A$$ wins.

• With $$n=10$$, the result is $$B$$, because $$A$$ can only choose $$1$$ or $$2$$ (because the number of digits of $$10$$ is $$2$$ and the sum of the digits of $$10$$ is $$1$$). So, when choosing, $$n$$ can only be $$8$$ or $$9$$. Next $$B$$ selects $$8$$ or $$9$$ and wins.

My problem is to determine who the winner is when $$n$$ is known.

• And how far have you gotten with it? May 4, 2019 at 8:41
• Try figuring out who wins if the the number is $11,12,13,\dots$ and see if can find a pattern. May 4, 2019 at 8:50
• If the number is $13$, A loses. Also, three examples are nowhere near enough. Actually, I don't see a pattern here. Are you supposed to find a formula or write a computer program? May 4, 2019 at 9:11
• I don't see a formula either. It seems very erratic. May 4, 2019 at 9:17
• Also posted to MO, mathoverflow.net/questions/330743/… with no notice to either site. THAT'S AN ABUSE! May 4, 2019 at 23:09