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Cut a cake

  1. Dec 14, 2008 #1
    There is a feast going on. 64 people are invited. A jumbo size cake is to be served which is 8ft x 8ft. However before cutting the cake a new guest comes on. So now 65 peoplw have to serve. How the cake should be cut into equal 65 squares (A square can have no more than 2 pieces, i.e. more than one cut). Each piece should be 1 square foot size.
  2. jcsd
  3. Dec 14, 2008 #2


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    Maybe I am missing something here, but how can you cut an 8 x 8 cake into squares with total area 65?
  4. Dec 15, 2008 #3
    I would say cut the cake into 64 1 foot square pieces and then take 1 of those pieces and cut it in half thickness-wise. You'll then 65 pieces that are all equally 1 foot square; they just won't all be the same thickness; but that wasn't a stipulation.
  5. Dec 15, 2008 #4


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    OK, if you start making extra assumptions you can.
    We cut the cake into uncountably many pieces and use Banach-Tarski to re-assemble it so as to get two cakes of the original size, thus having 63 pieces left after serving all the guests.
  6. Dec 15, 2008 #5
    I am afraid I was not able to form the quize well.

    3rd dimension is not included in the puzzle. Just consider it a flat piece (May be a bread).

    Banach-Tarski requieres a lot of cuts, I suppose.
    Please note: A square can have no more than 2 pieces, i.e. no more than one cut.

    Though it deals with a paradox.
  7. Dec 15, 2008 #6
    This still doesn't answer the most basic question raised:

    In your question, you stipulate that each piece should be 1 square foot in size. And you say that there should be 65 pieces. This means there MUST be a total of 65 square feet. However, you state that the cake is 8 feet x 8 feet, meaning that there are only 64 square feet of cake available!

    Therefore the answer is that they should go out and purchase 1 extra square foot of cake.

    But beyond that, you state something that I simply don't understand-- you say "A square can have no more than 2 pieces". What does that mean? You say "no more than one cut", but that also makes no sense. By definition, in order to create a square, you must spend at a minimum 2 cuts, and probably (for most pieces) a minimum of 4 cuts, since each piece is a square. Each edge of the square either needs to be cut from the cake (IE 1 cut per edge), OR an edge needs to use an existing edge of the cake (in which case its opposite edge would still need to be cut).

    You mention that there is a paradox involved, so perhaps we're meant to divide the cake using some mathematical paradox, like dividing by 0 or something. But the facts stand-- you CANNOT divide 64 square feet of cake into portions that equal 65 square feet.

  8. Dec 16, 2008 #7
    I think he means two equal right triangles can be reassembled back into a "square" with "one cut" splitting it. So no squares made from 4 cuts.
  9. Dec 17, 2008 #8
    I think the OP has in mind the puzzle in which a shape of area 64 is cut and reassembled to make a shape which seems to have an area of 65. Here is an example:
    If that is the case, then there are some issues with the OP's statement of the problem. While it is true that you might fool some of the people some of the time by showing them that upper quadrilateral and calling it a triangle. However, you cannot fool all of the people all of the time by handing out 65 pieces of 1 foot square cake.
    Last edited by a moderator: Apr 24, 2017
  10. Dec 17, 2008 #9


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    OK, but there is a difference between drawing a cake which looks like it has 64 squares but actually has 65 or vice versa (which is very much possible, according to jimmysnyder's link) and taking an object with a surface area of 64 and ask how to cut it into 65 pieces of area 1 (which is, by basic laws of arithmetic, impossible).
  11. Dec 19, 2008 #10
    You are very clever Jimmey. I found this puzzle in "Advanced Engg Mthematics by Erwin Kreyszig, 8th ed, P_Set 14.3, page 751". The answer has been found even without the diagrams. Fine!
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