Good Jokes and Great Problems

You may or may not agree with the following contention. (Typing that sentence made me think, “The law of the excluded middle either rules or does not rule.”)

A good joke is like a good math problem…
You don’t expect the punch line,
And you have to think to figure it out.

One problem for which this maxim holds is Paper Pool. The problem can be found in the unit Comparing and Scaling: Ratio, Proportion, and Percent, which appears in the middle-grades Connected Math Project curriculum. It also appears in 1000 Play Thinks by Ivan Moscovich as Playthink 833: Reflected Balls.

The rules for Paper Pool are rather straightforward. Create a rectangular pool table with integer dimensions that has a pocket at each of the four corners. Then following the rules below, in which pocket will the ball land, and how many hits will occur?

• The lower-left corner is always corner A, and the labeling continues counterclockwise with B, C, and D.
• The ball always starts in corner A.
• The ball is hit with an imaginary cue (a stick for hitting a pool ball) so that it travels at a 45° diagonal across the grid.
• If the ball hits a side of the table, it bounces off at a 45° angle and continues its travel.
• The ball continues to travel until it hits a pocket.

For example, a 5 × 3 table is shown below. Following the rules above, the ball will land in Pocket C after 8 hits. (Note that the initial strike by the cue stick at A and reaching the pocket at C are both counted as hits. You can disagree if you like, but using these conventions will make it easier to see some patterns.)

How many hits do you think will occur on a 5 × 4 table? Go ahead, take a second to think about it…

If you predicted that 9 hits would occur, you’re right:

You can now see a pattern begin to emerge:

5 × 3 → 8 hits
5 × 4 → 9 hits

So, then, how many hits will occur on a 5 × 5 table? Given the pattern above, it would be reasonable to think that 10 hits would occur. But that would be wrong…

Like the punch line to a good joke, this result was unexpected. At this point, most people want to figure out what’s going on.

If you’re like most people, then you can explore Paper Pool using the Online Paper Pool Tool at Illuminations.

Finally, here’s a joke that really does have an unexpected punch line.

Will you be able to attend the 2012 International Convention of the Barbershop Harmony Society in Portland?  You really should try to make it — it’s bound to be a harmonic function. 

Ouch.