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 Question from kelen, a student: a 10 inch by 10 inch square cake must be divided evenly among 5 people. the top and all four sides of the cake are frosted. each person must receive the same amount of cake and the same amount of frosting. there must be no cake left over. how can this be done?

Kelen,

Here's a hint: Assuming vertical cuts, the icing on the top of the cake will be divided fairly if the cake itself is. Therefore if the cake and the icing on the sides are both divided fairly you're done.

You will probably need to know the formulae for the areas of a triangle (1/2 base x height) and a trapezoid (average of parallel sides x distance between them).

Note that solutions with vertical cuts don't depend on the height of the cake, which is just as well because we don't know it. If you allow slanting cuts, there are solutions that do depend upon how tall the cake is. For instance, if the cake was a cube 10 inches high, frosted on five faces, one slice could be a pyramid going from the entire top to a point 4" above the center of the base. Then the rest could be cut symmetrically in quarters. But this would not work on a flatter cake; the first person would get most of the frosting.

Good Hunting!
-RD

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