HMMT 二月 2000 · ADV 赛 · 第 10 题

英文原题

10. I all t w o p eople A and B and think of a natural number n . Then I giv e the n um b ern to A and the number n + 1 to B. I tell them that they have b oth b een giv en naturaln 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 number is and vi e v ersa. I start b y asking A if he kno ws B's number. Hesa ys \no". Then I ask B if he kno ws A's number, 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 each 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 hesa y \no" b efore this? Assume that b oth A and B are v ery in telligen t and logi al. Y ouma y need to onsider m ultiple ases.