Problem #2: Handshakes

Problem 2. Handshakes

A man and his partner invite five other couples to dinner. As they enter the house, and accept pre-dinner drinks, the hosts and their guests shake hands with each other. Not all of them do, for various irrelevant reasons. However, naturally, no guest shakes hands with his or her partner. At dinner, the host brings up the subject of the handshakes. Going round the table and asking everybody else in turn how many hands he or she shook, he finds that he obtains 11 different answers. How many hands did his partner shake?

At first I was very overwhelmed with this problem. I had never seen one like this and didn't really know how to start. I knew that the maximum amount of shakes had tobe 10. I also didn't really read the problem entirely and thought I needed to find the

total number of hands that were shaken... so that's what I did first.

I used some pictures to help me figure out the number of handshakes for 0 to 6 people and then found the pattern. I subtracted 6 of the 66 handshakes that would have been for each other's partners to get 60 total handshakes and then I re-read the problem.

Once I realized what the problem was asking I started to just test out some possibilities. I drew dots and started drawing lines. I failed twice, first I started to go around the circle, person one shook one hand person two shook 2 and so on.

It didn't work out, so I tried skipping around and that didn't work either.

Eventually I tried to come up with some sort of plan. I started with the couple where one person shook 10 hands and the other didn't shake any and kind of ran with that. I started out with one person from every couple shaking every one's hands and this ended up working out. See Photos above

The wife of the host shook 5 peoples hands (and so did the host!)