Wednesday, April 6, 2011

How Many Handshakes?

The previous discussion reminds me of a radio show that the Head of School at Natural Friends did last summer.  As an example of the kind of deep problem that kids at Natural Friends might work on, he offered, "there are three people at a party.  How many handshakes?"

I was completely baffled by this the first time I heard it.  How the heck would I know how many handshakes?  There are too many possibilities for a divergent thinker like myself.

There might be zero handshakes, because everyone in the room is germ-phobic.

There  might be hundreds of handshakes, because it's a meeting of the Obsessive-Compulsive Handshakers Support Group.

If two of the three people are married to each other, they wouldn't need to shake hands, so there would be two handshakes (the husband and wife each shaking the third person's hand.)

If two of the three people are Orthodox Jewish men (not allowed to touch a woman other than their own wife), and the third is an unrelated woman, the only legal handshake would be between the two men.

Or it could be that each person shakes every other person's hand exactly once, in which case you have three handshakes. 

That last possibility is actually the preferred one; the handshake question turns out to be a classic problem.  For the problem to work as intended, it must be assumed (or better yet, stated) that each person shakes every other person's hand exactly once.

The problem is fairly sophisticated; as you add people to the party, you wind up taking the sum of an arithmetic series.  I think it would go right over the head of a kid brought up on the thin gruel of Trailblazers Math.


  1. It's a PARTY. Festive atmosphere. No handshakes. Two hugs, twenty, or zero... depending on how much alcohol is served. Unless... well, all that stuff you mentioned.

  2. Wow, I want to go to your party!

    I think the party for this problem is more like a boring political fundraiser.

  3. And is there maybe a little gender bias in there, too? If the three people were female, would we assume that they would all shake hands with each other at a party? (And it wasn't that long ago when men were instructed never to initiate a handshake with a woman.)

  4. Chris, I've never heard that men weren't supposed to initiate handshakes with women. What was that about?

    I agree, though, that handshakes have a basically masculine character. Shaking a woman's hand is a way of treating her as an "honorary man". That's why it's done in a business context.

  5. Yeah, that was the old rule. I think men just had no idea what they were supposed to do with women in any kind of business setting, so the idea was to take your cue from the woman.

    There are three people at a party. How many kisses on the cheeks? I bet even your Head of School would admit that the answer to that question is "It depends."

  6. Plus, you don't usually give problems to schoolkids that involve kissing.

  7. I just discovered that the Handshake Problem is mentioned in the Connected Math curriculum. Here's a good article about Connected Math, from Barry Garelick:

    What is Discovery Learning?

  8. I think the party for this problem is more like a boring political fundraiser.