Solution to: Happy Handshaking

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.

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