Dividing A Cake - A Math PuzzleDate: 12/20/98 at 19:26:16 From: jim Subject: Dimensions/algebra/cake cutting You have a nine-inch-square cake. It is a two-layer cake. Each layer is one and one-half inches high. Each layer is covered with 1/4-inch-thick frosting and so are the sides. There is no frosting on the bottom. Using straight knife cuts, how can you divide the cake into 13 pieces so that each piece has exactly the same amount of cake and frosting? After pieces are cut they may be put back together, but you may not remove any cake or frosting. Date: 12/21/98 at 13:23:03 From: Doctor Wilkinson Subject: Re: Dimensions/algebra/cake cutting This is a very nice puzzle. In case you want to work on it a little longer on your own, I'll give you a couple of hints. If you get stuck or you just want the answer, let me know. 13 is too big; it's hard to visualize the problem with such a large number of pieces. Try some smaller number like 5 (4 or 2 would be too easy). If the cake were circular instead of square, it would certainly be easy. You would just mark off points on the circumference that would divide it into 5 equal parts, and cut from the center of the circle to these points, right? Now see if you can do something similar with a square cake. - Doctor Wilkinson, The Math Forum http://mathforum.org/dr.math/ Date: 12/21/98 at 20:38:11 From: Anonymous Subject: Re: Dimensions/algebra/cake cutting Thank you very much for your response. These hints are very helpful. I think that I know how to go about solving the problem, but could you please send me an explained solution? Thank you very much, Jim Date: 12/23/98 at 11:21:37 From: Doctor Wilkinson Subject: Re: Dimensions/algebra/cake cutting The hint I gave you was really a very good one. You just need to be bold and follow it exactly. If you divide the circumference of the cake into 13 equal parts and then make your cuts from the center of the cake to each of the 13 points on the circumference, all the pieces will have equal amounts of both cake and frosting! To see this, notice first that the amount of cake in a piece is just the area of the top of the piece times the thickness of the cake. The amount of frosting from the top is just the area of the top times the thickness of the frosting, and the amount of frosting from the side is just the length of the part of the piece on the edge of the cake times the thickness of the cake times the thickness of the frosting. Now if you divide the circumference into 13 equal parts, you've taken care of the frosting on the side, and you just need to be sure that the pieces have equal area to take care of the cake and the frosting on the top. There are two kinds of pieces of cake if you divide it as I suggested: triangular wedges, and pieces that go around a corner of the cake. For the triangular pieces, the area is one half the base times the height. The height is just the distance from the center of the cake to the edge, which is the same for all the pieces, and the base is one of those equal divisions of the circumference, so that's the same for all the triangular pieces also. I'll let you think about the case of the around-the-corner pieces, but they work too. Here's a picture: - Doctors Wilkinson and Sarah, The Math Forum http://mathforum.org/dr.math/ |
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