I follow several Mathematics twitter accounts, mostly so that I can run across some interesting points I didn’t know about and feel a little dumber the rest of the day (*oh, good grief, of ***course** if f is a quasi-convex function and y a convex combination of x and z then f(y) is less than or equal to the maximum of f(x) and f(z)). Mostly they’re little “huh” bits. Unfortunately I’ve lost which one I found this item from originally, but it was just a link to an interesting puzzle result: how to cut a cake into four equal pieces using a single slice.

The answer is at the Quantumaniac’s Tumblr page, and it’s a sweet solution done in pictures. The proof that the four pieces have equal area isn’t complicated, and could almost be done wordlessly, though it’s not in this case. If you want a spot of fun, try making the argument after looking at the cake picture on top of the page and without looking at the fuller argument below.

As usual with this sort of puzzle you have to dig into unstated assumptions and figure where you can overthrow them. In this case, while the four slices do end up with equal area they don’t come out with the same shape. And you have to suppose that one could make a cut that’s curved uniformly at every point. This isn’t a problem if you’re slicing things in a mathematics puzzle, or if you can use a science fictional cutting laser, but these are a little outside the bounds of ordinary kitchen utensils.

Still I can imagine adapting the solution: if we suppose the kitchen to have (say) an eight-inch round cake pan, we can imagine it to have four-inch round cookie-style cutters and use them to make the needed slices. That wouldn’t be done in a single continuous cut, but it would use stuff that might actually be in the kitchen. Thus the mathematical solution can be approximated by the real world, or vice-versa, depending on your point of view.

Left unanswered: how about if you need six pieces? (I don’t know whether there’s any one-slice way to do that.)

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## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.
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I assume one of the assumptions is that the cutting instrument can’t leave the thing being cut? Because I mean, if you can it’s trivial – just draw a really big lowercase alpha with the cross centered in the center of the cake and the loopy parts outside of it. (The generalization to any even number of slices is obvious.)

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To me, this doesn’t feel like a single ‘cut’ or ‘slicing’, though I suppose there’s a a mathematical way to construct it; while it is a continuous line it self-intersects in fashion that to me suggests it’s at least two cuts. I would hold that it’s a wonderful solution, but I’m inclined to quibble on defining this is one cut. It enters and then exits the cake at opposite point, then a second cut re-renters. However, mathematically, we’re talking tangent curves, so ‘exiting’ the cake isn’t quite true; but the cut hits the edge of the cake– and you’ve now reached the edge of the cake, at a different point that you began the cut at the edge of the cake. That’s where the first slice to me feels ‘done’, and it’s why I’m going to quibble here.

Also, if tangent cuts count, you can also *theoretically* achieve any number of equal-sized slices more than 2, by making ever-skinnier ever-more-legged asterisks centered in the middle of the cake, with the tips tangent to the edges. Calculating how precisely to figure the angles so each outer slice’s area would be equal to the area of the middle asterisk is not trivial, though.

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Well, yeah, part of the fun of an open-ended problem like this is figuring out what you do mean by a slice, or a continuous slice.

My gut instinct is to say that drawing such an alpha shape that goes outside the edge of the cake breaks the single-continuity of the slice. I can’t quite say that you have to stay in the interior, since the double-circle slice shown here touches the edge of the cake and therefore isn’t on the exterior. That implies allowing cuts that go to the edge but not extending past it. But then that opens up the exception: why not make the alpha slice, only instead of going outside the limits of the cake simply trace around the edge?

To say the cake slice can only touch the edge twice doesn’t seem quite right, since I can imagine trying to cut a slice by a trefoil or some other figure, easy to describe in polar coordinates, that might touch the edge many times over and still, well, look like a single if very adept bit of slicing work.

(I can think of a mathematically precise way to say what I want: that the set of points where the slice touches the exterior boundary of the cake should have measure zero, but that’s a couple hundred words to explain to anyone who hasn’t hit measure theory.)

Anyway, if you are willing to take the idea of a continuous slice that goes beyond the boundary of the cake, then, yeah, the alpha-shape slice is a workable answer. It does come to figuring out what the assumptions about a legitimate slice and, for that matter, equality are. Captain Packrat noted that if you have a layer cake, you can just cut the whole cake in half as a normal halving, and then separate the layers. That’s a legitimate answer too, if you’re just looking for cake volume and figure that since the kind of cake hadn’t been specified you can make that part of the answer.

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