Puzzle: slicing up a pie

image by Paul Smith

Holiday parties are one of the few times that I enjoy eating pies.

This is a fun and easy problem about the mathematics of slicing a pie:

Alice and Bob are preparing for a holiday party, and each has a pie to slice up into pieces.

They decide to have a little contest to make things fun. Each person is allowed to make 3 cuts of the pie with a knife, and whoever ends up with more pieces is the winner. They agree stacking is not allowed, but that “center” pieces without the crust are permissible.

How many pieces can be made using 3 cuts? What about 4 cuts, or more generally n cuts?

Can you solve it?

The answer is in the comments section.



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  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    Answer to the puzzle

    When you make one cut, you can create 2 halves. With two cuts, you can slice through each half again, to make 4 pieces.

    The third cut is a bit trickier. What you want is to cross the previous cuts without going through the intersection point. Every time you intersect a previous cut you create another section (piece) of the pie. So with 3 cuts, you can make 7 pieces in all:

    image credit: Wikipedia, author ArnoldReinhold, CC BY-SA 3.0

    We can now generalize. The first cut makes the pie into 2 pieces. But after that, making the cut n will add on n new pieces to the pie. Thus, we know that on cut n the total number of pieces will be:

    f(n) = 2 + 2 + 3 + 4 + … + n = 1 + (1 + 2 + 3 + …) = 1 + n(n+1)/2

    The number of cuts you can make with n cuts is known as lazy caterer’s sequence.

  • James Clare

    Technically with 3 cuts you can actually get 8 pieces of cake, two cuts on top you get 4 pieces and 3rd cut through the whole of the cake means you get 8 pieces, although i guess it depends on the thickness of the cake your cutting up.

  • Adnan

    @James – I like that logic for cake. Too bad a pie would not be so amenable. Is there a way to generalize that?

  • James Clare

    I think to genralize it is just 2^n.

    1-2
    2-4
    3-8
    4-16
    ….
    n-2^n

  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    There is a 3d generalization called “cake numbers”:

    http://en.wikipedia.org/wiki/Cake_number

    The cake number is the maximum number of regions that 3-dimensional cube can be divided using planes (or slices). The cake numbers are 1, 2, 4, 8, 15, 26, 42, 64, 93, …, 1/6 * (n^3 + 5n + 6)

  • Ben Sauer

    In addition to disallowing stacking, you should have disallowed rearranging. By making the pieces collinear you can slice them all in half with one cut. (For some definition of one cut.

  • john

    You also forgot to disallow curved cuts (I don’t think there is any definition of a cut that requires it to be straight). Otherwise, the answer would again be 2^n.

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