Logic Problem 2

[ericms]

Inspired by Dan's recent thread. I believe this one to be less of a pain in the ass, but probably more difficult. Also there is no possible way we can get into some ridiculous argument of trivialities or physics. I've simplified it to the case n = 2. Show this in a spoiler tag or something if you get it (I'd very much appreciate a proof along with an answer).

If the host of a party invites 2 couples, and if no one shakes hands with his or her partner, and if each of the 5 people (all couples plus the hostess) interrogated by the host shook a different number of hands, then how many hands did the hostess shake?

 

[Dan]

Why is the host interrogating his guests and his wife? Doesn't sound like a fun party. But the hostess shakes 2 hands:

4 is the max that anyone can shake without double hand shakes and all the social awkwardness that it would bring to the interrogation themed party. So the guests 4 guests plus hostess have to shake 0,1,2,3,4 hands. Whoever shakes 0 is completely out of the game, so that leaves 3 guests and the hostess with 1,2,3,4 handshakes. The partner of the person with 0 handshakes has to shake 4 hands (because they are the only one who can). So that is one couple satisfied. Then one member of the remaining couple has to shake 2 more hands to make 3 because they are the only one who can while leaving their partner with 1 handshake. This leaves the hostess with 2 handshakes.

 

[ericms]

Nice. The cool thing is that this generalizes to n. I had that as the problem before but setting it to 2 seemed like a better idea since it more or less involved the same argument.