Question

Two players A and B play the following game. A selects an integer from 1 to 10, inclusive of both. B then adds any positive integer from 1 to 10, both inclusive, to the number selected by A. The player who reaches 46 first wins the game. If the game is played properly, A may win the game if:

• A. A selects 8 to begin with
• B. A selects 2 to begin with
• C. A selects any number greater than 5
• D. None of the above

Hint:

Work backward from 46 in steps of 11 to find the checkpoint numbers 2, 13, 24, 35, 46, and see which starting number lets A control every checkpoint.

Answer 1

The Correct Option is B

This game is really about controlling fixed checkpoints. Since each move is a whole number from 1 to 10, a player can always reply to the opponent's move with $11 - m$ to make any round add up to exactly 11. Working back from the target 46 in steps of 11 gives the checkpoints 46, 35, 24, 13 and 2. Whoever occupies 2 right at the start (since A gets to freely pick the opening number here) can force every later checkpoint and win at 46. Let's test each option against this idea.

1. A selects 8 to begin with: 8 is not a checkpoint (the checkpoints are 2, 13, 24, 35, 46). If A opens with 8, B can add 5 to reach 13, a checkpoint, and from there B controls the rest of the game the same way A would have. So opening with 8 hands control to B, not A.
2. A selects 2 to begin with: 2 is exactly the first checkpoint. Whatever B adds, A can always add the complement to 11 and land on 13, then 24, then 35, then 46. This guarantees A the win no matter how B plays.
3. A selects any number greater than 5: this covers 6, 7, 8, 9 and 10, none of which are checkpoints. For every one of these, B can respond by landing on 13 and take over control, so this choice does not guarantee A a win.
4. None of the above: this would only be right if no starting number worked for A, but option 2 (selecting 2) does work, so this option is ruled out.

The only opening move that locks in a forced win for A is 2, since it matches the first checkpoint in the sequence 2, 13, 24, 35, 46.

Let's summarize:

• Whenever both players can add 1 to 10, a player can force the round total to rise by exactly 11 by replying with $11-m$.
• Working backward from the target 46 in steps of 11 gives the checkpoints 2, 13, 24, 35, 46.
• Whoever reaches a checkpoint first (here, A, by choosing the opening number) can always force the next one, and eventually the target itself.

So A should begin with 2 to guarantee a win, which is option (B).