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Birthday cake: Two fiends A, B are going to share a circular cake, as follows: They alternate turns doing a straight cut each, starting by A and doing 4 cuts in total. Then they take turns picking one piece at a time, again starting from A. Can you find a way so that B picks the pieces with minimum half the area of the cake (in total)?

My idea would be for B to imitate A's cuts, so as to create pieces of equal area in pairs. For example, I am thinking: 1st cut: any random chord, which splits the circle into two parts of area P and Q. Then B draws the vertical bisector of the chord, so he splits each part into two smaller parts of area P1=P2 and Q1=Q2. Then I guess we draw some other chord by a 3rd cut and the 4th cut must somehow split all the smaller pieces into two parts each. Is it possible? Then I guess it is easy for B to pick exactly the same parts that A picks.

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Why not cut with 4 diameters, at angles $0, \pi/4, \pi/2, 3\pi/4$? That gives you $8$ (which is even) identical slices of cake.

Example: enter image description here

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  • $\begingroup$ @Alex.vollenga: Then you should ask that as separate question, since it deserves a separate answer. $\endgroup$ – quasi Oct 9 '17 at 12:15
  • $\begingroup$ @quasi is right. Changing the question after you've asked it is considered kind of rude, because in doing so, you waste the time of folks who take the trouble to answer the original question. (By the way, the answer to your second question is also easy... you just need to think about how bad a random cut could be.) $\endgroup$ – John Hughes Oct 9 '17 at 12:22
  • $\begingroup$ I started to post the question I suggested, but I realized it had a trivial answer, so I deleted it. If I can find an interesting, non-trivial restatement, I'll edit it an undelete it. In the meantime, feel free to ask your own version of the intended question. $\endgroup$ – quasi Oct 9 '17 at 13:09
  • $\begingroup$ @John Hughes: Nice cake! (+1). Did you make it yourself (with some application), or is it a stock photo? $\endgroup$ – quasi Oct 9 '17 at 13:12
  • $\begingroup$ The usual thing is just to post a new question -- changing the wording of the existing question makes all the current answers look like nonsense, and that's not nice. Asking a new question is completely OK. A more common sequence is that you write a question, someone makes a comment saying "Isn't this a trivial solution?", and you answer saying "yes, I forgot to include that", and edit the question before anyone's taken time answering. Comments are exactly intended as a chance for folks to help you make your question clear. In this case, I thought it was clear, or I'd have asked about it. $\endgroup$ – John Hughes Oct 9 '17 at 13:39

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