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You have baked a rectangular sheet cake for your two brothers, and are about to cut it and serve it to them when the dog eats a rectangular portion of the cake. Now the cake looks a little like this:The cake

Your job now is to cut the cake so that each brother gets the same amount of cake. This is of course a three dimensional cake, but cutting the cake parallel to the top is not allowed, because each brother need to get the same amount of frosting. You are given only string, and have no measuring tools, though you can mark points on the outside of the cake.

Usually I give some ideas on how I might solve this problem, but for this particular one, I have very few. I've given some variables to the sides, and figured out how much cake each brother is supposed to get, but I can't for the life of me figure out how to cut it. Any help would be appreciated.

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    $\begingroup$ Do both brothers also have to get the same amount of dog-drool-covered perimeter? :) $\endgroup$ – Rahul May 29 '14 at 20:58
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    $\begingroup$ Yes, it must be evenly distributed. Otherwise they wouldn't be equal, and the person with less drool would feel jealous! $\endgroup$ – recursive recursion May 29 '14 at 23:06
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Draw a straight line through the centre of the original cake and the centre of the (missing) rectangle.

This divides the original cake into two parts of equal area, and the doggie piece into two parts of equal area. So it divides the mutilated cake into two pieces of equal area.

Remark: If we do not want to deal with the ghost of departed cake, the mutilated cake divides naturally into two rectangular pieces, and we can use the same idea.

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  • $\begingroup$ you are only permitted one cut through the cake. $\endgroup$ – recursive recursion May 29 '14 at 20:54
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    $\begingroup$ I have described the (single) cut. It goes through the two points described. If you want to join points of the cake, the point on the outside of the cake where the line meets the cake can be found with string. $\endgroup$ – André Nicolas May 29 '14 at 20:56
  • $\begingroup$ Oh, you're right. Sorry about that! $\endgroup$ – recursive recursion May 29 '14 at 21:07

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