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Fair slices (Posted on 2018-01-16) Difficulty: 2 of 5
The 3 kids are going to split a small square cake. It would be easy enough to cut it fairly into thirds but there is a catch:
It is frosted on the top and four sides and they each want to make sure they get a fair share of cake and frosting.

Find one or more simple (or complicated, creative, etc) ways to accomplish the task.

I offered to slice it into fourths and take one as a 'slicing fee' but that didn't go over well.

No Solution Yet Submitted by Jer    
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re: cake plus frosting | Comment 5 of 8 |
(In reply to cake plus frosting by xdog)

xdog's method with straight lines going to the center works fine.  

However, the conjecture about congruent cuts does not work in general.  The lines to the center are not all the same length, so (for instance) a single arc along the longest length will add more to one of the pieces than what it gains from an adjacent piece, if all the arcs are congruent.

What does work instead of a straight line to the center is any curve that has as much area on one side of the straight line as it has on the other.  For instance, an even number of alternating arcs of the same size, or a zig-zag with the same number of zigs and zags of the same size.  And these do not even need to be congruent.  You could have one straight line to the center, and one cut with equal alternating arcs, and one cut with matching (offsetting) zigs and zags.

  Posted by Steve Herman on 2018-01-17 08:54:10
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