Alice and Bill threw a party and invited four other couples. As each couple arrived there were greetings, including handshakes.

Later in the evening, Bill asked everyone, including Alice how many people they shook hand with. Every answer was different. No one shook hands with his or her own partner.

How many hands did Alice shake?

Let's treat this as a problem in math, not in etiquette.

Since each of 9 people shook a different number of hands, there are only 8 people eligible to shake hands with someone (exlude yourself and your spouse), then these 9 people shook 0,1,2,...,7,8 hands respectively.

Now consider the person who shook 8 hands. He didnt' shake his own, and he didn't shake his spouse's, but he did shake everyone else's. Therefore, the only person who shook 0 hands is the spouse of the one who shook 8 hands.

Now consider the person who shook 7 hands. He shook the hand of the one who shook 8. He didn't shake the hand of the one who shook 0. He shook 6 others, which has to be all of the remaining people except his own spouse. Since everyone but his own spouse has now shaken at least two hands, his spouse is the one who shook one hand.

Continuing this chain of logic, we see the that number of hands shaken by each pair of spouses totals 8. Since we know that everyone's number except for Bill's is unique, then it only works if Alice and Bill each shook 4 hands.

Posted by Jim Lyon on 2002-06-20 08:35:07