Hosted by The Math Forum## Problem of the Week 1101## It's a Piece of CakeMacPOW Home || Math Forum POWs || Search MacPOW
## SolutionThe correct answer is 4, which was found by Witold Jarnicki, Sam VanderVelde, Lyle Ramshaw, John Duncan, John Sullivan, Tim Boykett, Aaron Atlee, Terry Stickels, Philip Roe, and Piotr Zelinski. You might want to skip immediately to the diagram if you did not solve it. Thanks to all the correspondents on this. If looked at properly the first time, the construction is perhaps not all that interesting. But almost no one looks at it properly the first time. The score among respondents on the list is something like 19 to 10 in favor of the answer 720. The correct answer is 4. The answer for 1 radian is 84; most will say that the icing never returns to the top in this case because π is irrational. I suspect that the score would even be more lopsided in an experiment where no special encouragement or hint of difficulty was given. Try it on your colleagues with a hintless delivery. They will all say "720." Here are comments by Peter Winkler (Dartmouth) re. the source:
This wonderful puzzle was passed to me by French graduate student Thierry Mora, who heard it from his prep-school teacher Thomas Lafforgue. The puzzle (of whose origin Lafforgue is unsure) actually involved a second angle as well, indicating the amount of cake passed over between wedges; it STILL requires only finitely many operations to get all the icing back on top, as ambitious readers will verify.
The diagram below shows how four moves do the job. I published a demo showing all the moves in the general case (user inputs the angle, which can be symbolic or numeric) at http://demonstrations.wolfram.com/TheCakeIcingPuzzle/. The general formula for the period is 2k(k - 1) where k is Ceiling[2π Here is a comment from Matt Hancher of NASA Ames Research Center:
Thanks for the great problem! The answer is 4 whenever the angle is 180 + n, where n is between 0 and 180. And from Ross McConnell, Colo. State Univ.
It takes only four flips! Instead of being blinded by the coordinates around the circle, let's look at a frame of reference that starts at the beginning of the last slice we flipped. Here is a complete solution to the general problem by Witold Jarnicki (Poland).
Let r be the fraction of the cake corresponding to the angle Here is the diagram showing the solution, using 185° instead of 181° for clarity. The answer is 4 for any angle between 180° and 360°.
© Copyright 2008 Stan Wagon. Reproduced with permission. |

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9 September 2008