Halmos Handshare Problem As is common, academics will occasionally attend dinner parties. Halmos and his wife attended such a dinner party along with four other couples. During the cocktail hour, some of those present shook hands, but in an unsystematic way, with no attempt to shake everyone’s hand. Of course, no one shook his or her own hand, no one shook hands with his or her spouse, and no one shook hands with the same person more than once. During dinner, Halmos asked each of the nine other people present (including his own wife), how many hands that person had shaken. Under the given conditions, the possible answers ranged from 0 to 8 hands shaken. Halmos noticed that each person gave a different answer: one person claimed not to have shaken anyone else’s hand, one person had shaken exactly one other person’s hand, one person had shaken exactly two hands, and so on, to the one person who claimed to have shaken hands with all the others present, except his or her spouse, that is, 8 handshakes in total. So, in summary, of the 10 people present, people gave answers from 0 to 8 hands shaken, i.e. one person had shaken 0 hands, another 1 hand, another 2 hands, another 3 hands, etc., up to 8 hands.
How many hands did Halmos’wife shake?