The frosting on the math puzzle

So one of the side effects of living with a math/computer geek is that often reading material I would never encounter on my own appears around the house. For instance, Communications of the ACM, which I found open to the last page, title “Puzzled: Circular Food.” Skimming this my attention was drawn to the second puzzle. Now I’m not really a math type by any stretch of the imagination* but even I can tell you that the following puzzle is unsolvable as written:

A cylindrical ice-cream cake with the most scrumptious chocolate frosting on top is sitting on a table. As an expert cake cutter, you choose an arbitrary angle x and proceed to cut one wedge after another, counterclockwise, around the cake, each of angle exactly x. However, each time you cut a wedge, you turn that piece upside-down and slide it back into the cake. This puts the frosting on the bottom at first, but as you work your way around and around the cake, the frosting comes back up to the top, then returns to the bottom, and so forth. Your mission is to prove that after some finite number of slices all the frosting will be back on top of the cake.

So what’s the problem? There is an assumption embedded in this problem that is patently not true. If you really were to flip a single piece of frosted cake upside down and then right side up again you could do that an infinite number of times and still not end up with all the frosting back on top. The one exception, perhaps, is if it had something like a rolled fondant frosting. However, I argue that the specification of a “most scrumptious chocolate frosting” rules out fondant. Besides, fondant doesn’t freeze well so it would be an odd choice for an ice-cream cake. I think the author here is assuming that frosting on an ice-cream cake is solid enough to resist squishing and sticking to the plate when flipped upside down. I think, though, that this demonstrates a complete misunderstanding of cake physics.

I would revise this problem as follows:

An unfrosted cylindrical two-layer cake with a bottom layer of white cake and a top layer of chocolate cake is sitting on the table….

Then substitute chocolate layer for frosting in the rest of the problem. The cake won’t be as tasty but at least the question works. Besides if you’re eating hypothetical math cakes you have bigger problems than the relative tastiness.

And yes, I know that the fact that I obsessed enough about this to figure out how to make the problem work makes me my own type of geek. I’m ok with that. And no, I still have no idea how to go about solving the actual problem.

*Note that I don’t mean I’m not a math type in comparison to the general population since I imagine a lot of people would conclude that, given that my work all involves statistics, I’m a math whiz. I mean I’m not a math type in comparison to B. and the many friends I have who might have some idea how to start thinking about these puzzles (in a way that doesn’t nitpick the puzzle itself).

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