The pound cake, the math enigma of “Le Monde” n° 68

People traveled from far and wide to taste the famous cakes from the Le Quatre-Quarts pastry shop. On its front, one could read in large gold letters "A good cake is a cake to share" and these words were much more than a motto: they were a geometric constraint. You could indeed buy all sorts of cakes there, with more or less strange shapes, but which all had the following property: they were omnisective, that is to say, it was possible to cut them into any number of pieces of the same size and shape. So, if you bought a cake without knowing precisely how many guests it would be shared among, you couldn't go wrong!

Of course, one could find in its windows traditional round tarts, which could be divided into triangular slices at will (figure A) or rectangular desserts that could be cut into as many parallel slices as necessary (figure B). But one could also observe stranger shapes, whose omnisectrity was far from obvious.

Could you cut the cakes shown in figures C and D into two, three, four or more equal pieces (in shape and size)?

One day, the pastry chef wondered if he could offer a cake in the shape of an equilateral triangle. After some research, he found a way to cut it into two, three, or four equal pieces, but he couldn't completely resolve the issue for any number of pieces. He then contacted a surveyor who was an expert on the subject. However, her answer left him even more confused.

Can you guess why?

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