Jack and his wife went to a party where four other married couples were present. Every person shook hands with everyone he or she was not acquainted with. When the handshaking was over, Jack asked everyone, including his own wife, how many hands they shook. To his surprise, Jack got nine different answers.

How many hands did Jack's wife shake?

Ans:

Because, obviously, no person shook hands with himself or herself, or with his or her partner, nobody shook hands with more than eight other people. Moreover, since nine people shook hands with different numbers of people, these numbers must be 0, 1, 2, 3, 4, 5, 6, 7, and 8.

The person who shook 8 hands, shook hands with all other persons (who therefore shook each at least 1 hand), except with his or her partner. Therefore, the partner of the person who shook 8 hands must be the person who shook 0 hands.

The person who shook 7 hands, shook hands with all other persons (who therefore shook each at least 2 hands), except with his or her partner and the person who shook 0 hands. Therefore, the partner of the person who shook 7 hands must be the person who shook 1 hand.

The person who shook 6 hands, shook hands with all other persons (who therefore shook each at least 3 hands), except with his or her partner and the persons who shook 1 and 0 hands. Therefore, the partner of the person who shook 6 hands must be the person who shook 2 hands.

The person who shook 5 hands, shook hands with all other persons (who therefore shook each at least 4 hands), except with his or her partner and the persons who shook 2, 1, and 0 hands. Therefore, the partner of the person who shook 5 hands must be the person who shook 3 hands.

The only person left, is the one who shook 4 hands, and which must be Jack's wife. The answer is: Jack's wife shook 4 hands.