BTGmoderatorDC wrote: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


OA D

Source: Official Guide

Let's go through the choices.

A. 7

Since Rita goes first, she can remove only 1 stick. (If she removes any other number of sticks, then Sam will be able to win on his next turn.) Then there are 6 sticks left, and no matter how many sticks Sam removes, Rita will always win (for example, if Sam removes 2 sticks, Rita can then remove 4 sticks for the win). Since we want Sam to win, choice A is not the correct answer.

B. 10

Since Rita goes first, she can remove 4 sticks. Then there are 6 sticks left, and no matter how many sticks (up to the 5-stick limit) Sam removes on his turn, Rita will win on her next turn. Thus, there is no way Sam can win.

C. 11

Since Rita goes first, she can remove 5 sticks. Then there are 6 sticks left, and just as in the explanation for answer choice B, there is no way Sam can win.

D. 12

Since Rita goes first, she can remove any number of sticks, from 1 to 5 (inclusive). Then Sam can remove a number of sticks so that there are 6 sticks left. For example, if Rita removes 2 sticks, Sam should remove 4 sticks so that there are 6 sticks lefts. Once there are 6 sticks left and since it's now Rita's turn, no matter how many sticks she removes, Sam will always win (for example, if Rita removes 5 sticks, Sam can then remove 1 stick for the win).

Answer: D