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E19. Cutting a square cake

I found the following puzzle in Jacob's Geometry: Seeing, Doing, Understanding (3rd ed.). Given a square sheet cake, 9" on a side, divide it into 5 pieces that have the same amount of cake and the same amount of icing.

To be more mathematically precise, the problem is: Given a square, a units on a side. find a dissection of the square into 5 polygonal pieces, each with area a2/5 and each containing the same length of boundary of the original square, namely 4a/5. I think this is not too hard, but I will post my answer if anyone asks.

Extra credit: To generalize and sharpen this result, show that we can replace 5 by n, where n is any integer greater than or equal to 3. Also show that all polygons can be triangles or convex quadrilaterals, and that even so for any fixed n there are an infinite number of essentially different such dissections. (Two dissections are essentially different if one contains a polygon that is not congruent to any polygon in the other dissection.)

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