Shake Hands

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?


Answer:


There were ten people at the party. No one shook hands with his or her own partner (and couldn't have shaken with him/herself) so the maximum handshakes a person could have made was 8. The minimum is 0. There are 9 numbers between 0 and 8 both inclusive, and 9 different answers, so each number was used as an answer once. The person who shook hands eight times shook with everyone but his partner, so none of them can be the person who did not shake at all, so she must be "8's"partner.
After eliminating "0" and "8", the same logic shows that "7" and "1" must be partners, then "6" and "2", followed by "5" and "3". "4's" partner must have also shaken 4 hands. But no two people answered the same. However, Bill did not answer the question, so he is the missing"4" Alice is his partner, so she is the other "4".

Additional explanation:

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.