My Mathematical Magic Show: Part 3a

Last March, on Pi Day (March 14, 2015), I put together a mathematical magic show for the Pi Day festivities at our local library, compiling various tricks that I teach to our future secondary teachers. I was expecting an audience of junior-high and high school students but ended up with an audience of elementary school students (and their parents). Still, I thought that this might be of general interest, and so I’ll present these tricks as well as the explanations for these tricks in this series. From start to finish, this mathematical magic show took me about 50-55 minutes to complete. None of the tricks in this routine are original to me; I learned each of these tricks from somebody else.

For my second trick, I’ll show something that my math teacher taught me when I was about 13 or 14. Everyone in the audience has a piece of paper and a pen or pencil. Here’s the patter:

Magician: Tell me a number between 5 and 10.

Child #1: (gives a number, call it $x$)

Magician: On a piece of paper, draw a shape with $x$ corners. Don’t draw something really, really tiny… make sure it’s big enough to see well.

Audience: (draws a figure; an example for $x=6$ is shown) The Magician also draws this figure on the board.

Important Note: For this trick to work, the original shape has to be convex… something shaped like an L or M won’t work. Also, I chose a maximum of 10 mostly for ease of drawing and counting (and, for later, calculating).

Magician: Tell me another number between 5 and 10.

Child #2: (gives a number, call it $y$)

Magician: Now draw that many dots inside of your shape.The Magician also draws $y$ dots inside the figure on the board, an example for $y = 7$ is shown.

Audience: (starts drawing $y$ dots inside the figure) The Magician also calculates $2y + x - 2$ and says, “Now while you’re doing that, I’m going to write a secret number on the board,” discreetly writes the answer on the board, and then covers up the answer with a piece of paper and some adhesive tape.

Magician: Now connect the dots with lines until you get all triangles. Just be sure that no two lines cross each other. For example, your figure could look like this:

Audience: (quietly connects the dots until the shape is divided into triangles)

Magician: Now count the number of triangles.

Audience: (counts the triangles)

The reason this magic trick works so well is that it’s so counter-intuitive. No matter what convex $x-$gon is drawn, no matter where the $y$ points are located, and no matter how lines are drawn to create triangles, there will always be $2y + x - 2$ triangles. For the example above, $2y+x-2 = 2\times 7 + 6 - 2 = 18$, and there are indeed $18$ triangles in the figure.