Consider a cylindrical cake, its top surface white and its bottom surface black. We will repeatedly cut out a slice of the cake, of angle theta, invert it, and replace it. The cake is made of icecream so the slice will immediately weld itself back seamlessly into place. The left edge of each slice will coincide with the right edge of the previous slice, so we will work round and round the cake.
After how many slices will the cake be entirely white on top again? Is this number necessarily finite?
This page links to a program which you can use to experiment with various angles.
Use the form to specify, in degrees, the size of the slices you want flipped. You should also change the "name" to something other than "default", otherwise your cake may get mixed up with someone else's.
Here is how the program's cake starts:
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