Geometric magic trick

This is a magic trick that my math teacher taught me when I was about 13 or 14. I’ve found that it’s a big hit when performed for grade-school children.

Magician: Tell me a number between 3 and 10.

Child: (gives a number, call it x)

Magician: On a piece of paper, draw a shape with x corners.

Child: (draws a figure; an example for x=6 is shown)

geom_magic1Important 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 3 and 10.

Child: (gives a number, call it y)

Magician: Now draw that many dots inside of your shape.

Child: (starts drawing y dots inside the figure; an example for y = 7While the child does this, the Magician calculates 2y + x - 2, writes the answer on a piece of paper, and turns the answer face down.

geom_magic2Magician: Now connect the dots with lines until you get all triangles. Just be sure that no two lines cross each other.

Child: (connects the dots until the shape is divided into triangles; an example is shown)

geom_magic3Magician: Now count the number of triangles.

Child: (counts the triangles)

Magician: Was your answer… (and turns the answer over)?

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.

Why does this magic trick work? I offer a thought bubble if you’d like to think about it before scrolling down to see the answer.

green_speech_bubbleThis trick works by counting the measures of all the angles in two different ways.

Method #1: If there are T triangles created, then the sum of the measures of the angles in each triangle is 180 degrees. So the sum of the measures of all of the angles must be 180 T degrees.

geom_magic4Method #2: The sum of the measures of the angles around each interior point is 360 degrees. Since there are y interior points, the sum of these angles is 360y degrees.

geom_magic5The measures of the remaining angles add up to the sum of the measures of the interior angles of a convex polygon with x sides. So the sum of these measures is 180(x-2) degrees.

geom_magic6In other words, it must be the case that

180T = 360y + 180(x-2), or T = 2y + x - 2.

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