HMMT 十一月 2021 · GEN 赛 · 第 1 题

HMMT November 2021 — GEN Round — Problem 1

A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated. a b b a 2 2

解析

A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated. Proposed by: Sean Li Answer: 4 Solution: Without loss of generality assume that the the left end of the first domino is red. Then, we have two cases: If the first domino is red-red, this forces the second domino to be red-blue. The third domino cannot be blue-red, since the fourth domino would then be forced to be blue-blue, which is impossible. However, RR RB BB BR works. If the first domino is red-blue, then the second domino cannot be blue-red, since otherwise there is nowhere for the blue-blue domino to go. Therefore, the second domino is blue-blue, which forces the third to be blue-red, and forces the fourth to the red-red. This yields one possibility. Therefore, if the first color is red, there are 2 possibilities. We multiply by 2 to yield 4 total possibilities. a b b a 2 2