Who wins the game?

Question

Two people play a game. The first one (A) picks one number from 1 to 10 and then the second one (B) also picks a number from 1 to 10 and adds it to the first number; the next round continues in the same way. Whoever gets to 45 is winner.

For example, A->4 and B->5 (4+5=9) A->3 (9+3=12) ... B->3(42+3=45) B is winner.

If A picks 3 in their first move, which number B can choose that definitely win the game for B?

1 2 3 4 5 6 7 8 9 10

Answer 1

9

Reasoning:

B's goal is to be able to answer any A's move with a move which guarantees B the victory. If B's first move is 9, the current sum is 12, and the sum of numbers left to 45 is 33, which is divisible by 11. After that whatever number x A picks, B answers with a number 11-x (which is always possible), increasing the current sum by 11 and reaching 45 on the fourth round.