Question 7

Step 1 · Analyze the First Move and Second Player's Response

The first player says a number $N_1$ between $1$ and $10$.

Since $1 \le N_1 \le 10$, the first player cannot say $11$.

The second player wants to reach the first winning number, $11$, by adding $k_2$: $N_1 + k_2 = 11 \implies k_2 = 11 - N_1$

Since $1 \le N_1 \le 10$, we have: $11 - 10 = 1 \le k_2 \le 11 - 1 = 10$

For example, if Player 1 says $5$, Player 2 adds: $11 - 5 = 6$

The total becomes $11$.

Step 2 · Maintain Control to Reach 99

On the next turn, Player 1 must add $k_3$ ($1 \le k_3 \le 10$). The new total is between: $11 + 1 = 12 \quad \text{and} \quad 11 + 10 = 21$

Player 2 then adds $k_4 = 22 - T$ (where $T$ is the current total) to reach $22$.

Continuing this strategy, Player 2 can always say every multiple of $11$: $11, 22, 33, 44, 55, 66, 77, 88$

When Player 2 reaches $88$, Player 1's next addition will result in a total $T$ between: $88 + 1 = 89 \quad \text{and} \quad 88 + 10 = 98$

Player 2 adds $k_y = 99 - T$ (which is between $1$ and $10$), reaches $99$, and wins the game.

Step 1 · Work Backward from the Target Number

To guarantee reaching $99$ on the final turn, the opponent must be forced to say a number $X$ such that: $99 - 10 = 89 \le X \le 99 - 1 = 98$

To force the opponent into the range $[89, 98]$, the winning player must say $88$.

With maximum addition $10$, the constant step size is: $10 + 1 = 11$

Subtracting $11$ iteratively backward: $88 - 11 = 77$

Continuing this pattern down to the start gives the sequence of winning numbers: $11, 22, 33, 44, 55, 66, 77, 88, 99$