HMMT 二月 2000 · ADV 赛 · 第 10 题

HMMT February 2000 — ADV Round — Problem 10

I all t w o p eople A and B and think of a natural n um b er n . Then I giv e the n um b er n to A and the n um b er n + 1 to B. I tell them that they ha v e b oth b een giv en natural n um b ers, and further that they are onse utiv e natural n um b ers. Ho w ev er, I don't tell A what B's n um b er is and vi e v ersa. I start b y asking A if he kno ws B's n um b er. He sa ys \no". Then I ask B if he kno ws A's n um b er, and he sa ys \no" to o. I go ba k to A and ask, and so on. A and B an b oth hear ea h other's resp onses. Do I ev er get a \y es" in resp onse? If so, who resp onds rst with \y es" and ho w man y times do es he sa y \no" b efore this? Assume that b oth A and B are v ery in telligen t and logi al. Y ou ma y need to onsider m ultiple ases.

解析

A will sa y y es when B sa ys no to n 1 or n , as A will then kno w B's n um b er is one n 1 greater than A's n um b er. Th us, A resp onds rst, after "no" resp onses if n is 2 n o dd , after "no" resp onses if n is ev en . 2