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?
Each couple is 'paired' such that there was a total of 8 handshakes between them. Thus, one couple had 0 and 8, one had 1 and 7, etc. One of the couples must have had 4 and 4. Since every answer was different, Bill must not have asked one of the couple's members how many hands he or she shook. Since he asked everyone, including Alice, the only person that could have been left out was himself. Since he shook 4 hands, Alice did as well.
For Alan
