握手次数
How many people did Mrs. Jones shake hands with?
题目详情
琼斯夫妇邀请另外四对夫妇来家里聚会（共 5 对夫妇、10 个人）。聚会结束时，琼斯先生问除他自己以外的每个人：你今晚和多少人握过手？并发现每个人给出的答案都不同。
当然：没有人会和自己的配偶握手，同一对人也不会重复握手。
问：琼斯夫人握了多少次手？
英文原题
Mr. and Mrs. Jones invite four other couples over for a party. At the end of the party, Mr. Jones asks everyone else how many people they shook hands with, and finds that everyone gives a different answer. Of course, no one shook hands with his or her spouse and no one shook the same person's hand twice. How many people did Mrs. Jones shake hands with?
解析
除琼斯先生外共有 9 个人，握手次数互不相同，且每个人最多只能与除配偶外的 8 人握手，所以这 9 个不同答案只能是
握 0 次的人，其配偶必然握了 8 次（与除自己配偶外的所有人握手）；同理，握 1 次者配偶握 7 次，握 2 次者配偶握 6 次，握 3 次者配偶握 5 次。
这就把 8 个数字两两配成 4 对配偶，剩下唯一未配对的数字是 4，对应的那个人只能是琼斯夫人（因为琼斯先生并未被问）。
因此琼斯夫人握了 次手。
英文解析
Excluding Mr. Jones, there are 9 people. The number of handshakes is distinct for each person, and each person can shake hands with at most 8 others (excluding their spouse). Therefore, the 9 distinct answers must be:
The person who shook 0 hands must have a spouse who shook 8 hands (shaking hands with everyone except their own spouse); similarly, the person who shook 1 hand has a spouse who shook 7 hands, the person who shook 2 hands has a spouse who shook 6 hands, and the person who shook 3 hands has a spouse who shook 5 hands.
This pairs up the 8 numbers into 4 couples of spouses, leaving the only unpaired number as 4. The person corresponding to this number must be Mrs. Jones (since Mr. Jones was not asked).
Therefore, Mrs. Jones shook hands.