[Feature photo above by Tobias Wolter (CC-BY-SA-3.0) via Wikimedia Commons.]
If seven people meet at a party, and each person shakes the hand of everyone else exactly once, how many handshakes are there in all?
In general, if n people meet and shake hands all around, how many handshakes will there be?
Our homeschool co-op held an end-of-semester assembly. Each class was supposed to demonstrate something they had learned. I threatened to hand out a ten question pop quiz on integer arithmetic, but instead my pre-algebra students presented this skit. You may adjust the script to fit the available number of players.
7 friends (non-speaking parts, adjust to fit your group)
Each friend will need a sheet of paper with a number written on it big and bold enough to be read by the audience. The numbers needed are 0, 1, 2, 3, … up to one less than the number of friends. Each friend keeps his paper in a pocket until needed.
[As the first narrator begins to speak, the friends come on stage and silently greet each other.]
Narrator #1: Seven friends meet each other in the hallway before class. Being polite young ladies and gentlemen, they want to shake hands with each other, and they don’t want to hurt anyone’s feelings or leave anyone out.
[Friends may give the narrator a strange look, but then they begin to shake hands all around.]
Narrator #1: If each friend shakes the hand of every one of the other friends exactly once, how many handshakes will there be?
[Narrator #1 takes guesses from the audience, which will almost surely be much larger than the actual answer. For best results, avoid taking guesses from any audience members who look like they are actually doing mental arithmetic. For instance, I refused to call on anyone from the MathCounts team.]
[Meanwhile, the friends on stage pantomime counting each other, counting on their fingers, and generally looking puzzled. As Narrator #2 begins speaking, the friends move to the side of the stage.]
Narrator #2: In our math class, we learned a way to organize this problem so that we can find the answer. We start with small numbers and work our way up. If there is only one person, there can’t be any handshakes.
[Friend with the “0” comes back to center stage, tries to shake hands with himself, shrugs, and then holds up his paper. He will continue to hold the “0” where it is visible for the rest of the skit.]
Narrator #2: If another friend comes along, there is one handshake.
[Friend “1” comes out, shakes hands with “0”, and holds up his paper. As the narrator continues, each friend in turn comes out and shakes the hands of everyone in turn, then holds up the number on his paper.]
Narrator #2: When the next friend comes along and shakes both their hands, that adds two more handshakes to the party…
The next friend adds three handshakes to the total…
The next friend adds four more…
The next friend adds five…
And the last friend shakes all six of their hands.
Narrator #3: Now, how can we add up all those handshakes to find the total number? The easiest way to add a long list of numbers is to rearrange them into tens.
[Friends on stage rearrange themselves into small groups as Narrator #3 continues.]
Narrator #3: We can see that 6 and 4 together are 10. Then 2, 3, and 5 make another ten. That is 20 so far, and 0 and 1 bring our total up to 21 handshakes in all.
[All players bow to the audience. Exeunt.]
We did not extend the skit to teach the general formula, but it is an interesting problem for beginning algebra students: If n people meet and shake hands all around, how many handshakes will there be? Can you figure it out?