Pizza Theorem

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Pick a point inside a circular pizza, and make two perpendicular slices through that point. Then make two more slices through this point that bisect the right angles made by the first two slices. The result will be eight pizza slices, each having a 45 degree angle at the point of intersection (see the picture). If you take every other slice, will the dark pie-shaped pieces give you more pizza, or the light pie-shaped pieces?

The Pizza Theorem says that the area of both regions is the same (regardless of where the initial point is placed)! How could you possibly prove this? This puzzle shows why (for this example) -- because each maple piece has a walnut piece of the same size and shape. With some extra thought you'll see that this dissection can be adapted to prove the theorem regardless of where the initial point is located.

The base of the puzzle is redwood from California, roughly 12 x 15 inches in size; the maple and walnut from my backyard in Norway Valley. The puzzle is fun to put together (there's more than one way to do it), but perhaps more significantly, the wood is so gorgeous it is a pleasure to have it on display as a work of art, as well as a conversation piece.

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