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!)

Problem #1: An equation

Problem 1. An Equation x² + y² = z²

After looking at the above equation, respond to the following: “What are some answers?

I immediately think about the Pythagorean theorem and finding lengths of the sides of a right triangle (especially after reading through problem #3). But because the variables are not a, b and c this leads me to believe that it might not be what this problem is getting at.

Answers could also be asking for solutions or numbers that make the equation true. You could come up with integer combinations (x = 3, y = 4 & z = 5 for example) or combinations that include irrational numbers (x = 1, y = 2 & z = sqrt 5). When I first was thinking about this I immediately drew these number combinations into triangles.

AFTER THINKING MORE:

I also am thinking about an equation of a circle (although the z should be an r). If we where to graph an equation like this it would be a circle centered at the origin with a radius of z. This answer took me a lot longer to come up with, I think because Pythagorean is so much more of a "major" mathematics theme. Even though I just got done with a unit on Conics right before Christmas, I literally just though of this as I was typing...