题目信息
Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4 or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?
A:7
B:10
C:11
D:12
E:16
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已选答案：
正确答案：
D:12
Arithmetic Elementary combinatorics
Let Player A be either Rita or Sam, and let Player B be the other player. If, after one of Player A's turns, there are exactly 6 sticks left, then Player A can win on his or her next turn. This is because if 6 sticks are left after Player A's turn, then regardless of whether Player B removes 1, 2, 3, 4, or 5 sticks, it follows that Player A can win on his or her next turn by removing, respectively, 5, 4, 3, 2, or 1 stick.
n = 7: If Rita begins by removing 1 stick, then there will be 6 sticks left after Rita's turn. Therefore, by the remarks above, Rita can win. Hence, Sam cannot always win.
n = 10: If Rita begins by removing 4 sticks, then there will be 6 sticks left after Rita's turn. Therefore, by the remarks above, Rita can win. Hence, Sam cannot always win.
n = 11: If Rita begins by removing 5 sticks, then there will be 6 sticks left after Rita's turn. Therefore, by the remarks above, Rita can win. Hence, Sam cannot always win.
n = 12: If Rita begins by removing 1 stick, then Sam can win by removing 5 sticks on his next turn, because 6 sticks will remain after Sam's turn. If Rita begins by removing 2 sticks, then Sam can win by removing 4 sticks on his next turn, because 6 sticks will remain after Sam's turn. By continuing in this manner, we see that if Rita begins by removing k sticks (where k is one of the numbers 1, 2, 3, 4, or 5), then Sam can win by removing (6−k) sticks on his next turn because 6 sticks will remain after Sam's turn. Therefore, no matter how many sticks Rita removes on her first turn, Sam can win by removing appropriate numbers of sticks on his next two turns. Hence, Sam can always win.
n = 16: If Rita removes 4 sticks on her first turn, then Sam will be in the same situation as Rita for n = 12 above, and therefore Rita can win no matter what Sam does. Hence, Sam cannot always win.
The correct answer is D.
Let Player A be either Rita or Sam, and let Player B be the other player. If, after one of Player A's turns, there are exactly 6 sticks left, then Player A can win on his or her next turn. This is because if 6 sticks are left after Player A's turn, then regardless of whether Player B removes 1, 2, 3, 4, or 5 sticks, it follows that Player A can win on his or her next turn by removing, respectively, 5, 4, 3, 2, or 1 stick.
n = 7: If Rita begins by removing 1 stick, then there will be 6 sticks left after Rita's turn. Therefore, by the remarks above, Rita can win. Hence, Sam cannot always win.
n = 10: If Rita begins by removing 4 sticks, then there will be 6 sticks left after Rita's turn. Therefore, by the remarks above, Rita can win. Hence, Sam cannot always win.
n = 11: If Rita begins by removing 5 sticks, then there will be 6 sticks left after Rita's turn. Therefore, by the remarks above, Rita can win. Hence, Sam cannot always win.
n = 12: If Rita begins by removing 1 stick, then Sam can win by removing 5 sticks on his next turn, because 6 sticks will remain after Sam's turn. If Rita begins by removing 2 sticks, then Sam can win by removing 4 sticks on his next turn, because 6 sticks will remain after Sam's turn. By continuing in this manner, we see that if Rita begins by removing k sticks (where k is one of the numbers 1, 2, 3, 4, or 5), then Sam can win by removing (6−k) sticks on his next turn because 6 sticks will remain after Sam's turn. Therefore, no matter how many sticks Rita removes on her first turn, Sam can win by removing appropriate numbers of sticks on his next two turns. Hence, Sam can always win.
n = 16: If Rita removes 4 sticks on her first turn, then Sam will be in the same situation as Rita for n = 12 above, and therefore Rita can win no matter what Sam does. Hence, Sam cannot always win.
The correct answer is D.
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