Sharing a sandwhich

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Posted · Report post

Given globs of ham, bread, and cheese (in any shape), placed any way you like, Prove that with a knife there exists a way to bisect each of the ham, bread, and cheese. In other words, show that you can share it with a friend so that each of you can have the exact same amount of the three globs of food.

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Posted · Report post

You mean show that for any shape there exists a direction such that if you slice through it you get two equal halves?

Well then just put the shape at point (0,0,0) and imagine slicing it parallel to the yz axis at point x, let v1(x) be the volume of the first half and v2(x) the volume of the second, if x is at the far left then v1(x)=0 and v2(x)=1, if x is at the far right then v1(x)=1 and v2(x)=0, all that's left to show is that v2 and v2 are continous and then simply apply the mean value theorem and you know there exists some x where v1(x)=v2(x)=0.6

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Posted · Report post

Spoiler

Identify the centerpoint of each of the 3 items and then slice a plane that passes through all 3 centerpoints. Since a line through the center point of each item bisects that item, you've cut the assemblage in half as well.

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