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blog.plover.com
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Very fascinating and interesting write-up. Thanks for sharing!

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I’m tangentially reminded of another lovely cake cutting problem (by way of Martin Gardner): there is a well known algorithm to decide a cake “fairly” between two people with no measuring devices - one person cuts and the other one chooses. “fairly” here implies that each party is satisfied that they got half the cake or more, regardless of the actual sizes of the pieces. now can you generalise this to N people? i.e. using no measuring devices, divide a cake among n people such that each person is satisfied they have gotten at least 1/N of the cake.

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The whole premise of the question is wrong, the “problem” of cutting the square cake into 9 pieces each with equal cake and frosting (which is on the sides and top) has a trivial solution.

Grandma could cut her cake diagonally, making four equal triangles. Then each triangle is cut into three subtriangles, each with equal base (side with frosting). Since they will all have the same height (not cake height, but base-to-vertex height), they thus have the same area, thus the same amount of cake and top-level frosting. And since the base sides are equal, they also have equal amounts of base-side frosting.

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That leaves 4 x 3 = 12 pieces, that cannot equally be divided into 9. Or am I missing something?

In any case, the situation you’re describing is just a special case of the one discussed in the linked post. .

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Then each triangle is cut into three subtriangles, each with equal base

Even just trisecting a line segment isn’t quite trivial, as a geometry problem. But on the other hand, the solution given seems to rely on measuring and evenly dividing the cake perimeter, so it’s not pure geometry.

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It’s still not that trivial. The nonzero thickness of the icing will mean the cake is no longer perfectly square on the corners, and even if you could keep it perfectly square (perhaps with a try square or cake-making jig), the ratio of icing to cake will vary around the perimeter (unless Grandma adjusts the icing thickness accordingly, but by the time I’ve explained that she’s switched to a circular pan).

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The source of the puzzles is a list with the title “Math puzzles for dinner”. It’s implicit that the icing has zero thickness but is still very desirable to eat.

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Now we’re off into spherical cow territory! I think it would still work with nonzero-thick icing if Grandma just doesn’t ice the corners or edges of the cake, so that the icing is just an extrusion of the cake surface. Grandma’s pretty good with icing, she can probably do that.