Quartering a Cake in One Slice


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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