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Piece o' Cake (Posted on 2003-12-04) Difficulty: 3 of 5
I was sitting down with Stefanie one day to share a round cake (our birthdays are only two weeks apart). "This is easy enough," I said, "one cut right through the middle divides the cake into two equal pieces."

Then, two more people showed up, but I was undaunted. Two straight cuts will divide the cake into four equal parts, I thought.

Then, I saw another car pulling up. I remembered that three straight lines can divide a circle into at most seven parts, but I was unsure if that could be done so that all the pieces are equal (in volume, not necessarily in shape).

How can I use three straight cuts to divide our cake into all equal parts and accomodate the greatest number of people?

Note: since Stefanie spent so much time decorating the cake, I don't want to rearrange the pieces when I cut them.

See The Solution Submitted by DJ    
Rating: 3.6667 (9 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution re(2): define... | Comment 17 of 22 |
(In reply to re: define... by DJ)

I don't remember seeing that there before. I wonder if my mind just skipped over it.

In that case, making two perpendicular vertical slices and a horizontal slice will do the trick. For a more creative solution, there might be a way to have three slanted slices and have 8 equal volume pieces. I think it's impossible to have more than 8.

I propose that there is a way to have 3 slanted slices go through the cake so that there is a triangle on top and a backwards triangle on bottom. This would form 8 pieces, 6 on the sides, and 2 pyramids on the top and bottom. Since it is symmetrical, all side pieces already have equal volumes. The center pyramids would each have a volume of 1/3*H/2*S²*√3/4, where S is the side of the triangle base and H is the height of the cake. This must equal 1/8 the volume of the cake, which is πR²*H, with R the radius.

1/3*H/2*S²*√3/4=πR²*H/8
S²*√3/3=πR²
S²=√3*π*R²
S=√(√3*π)*R

The distance from the center to the base triangle's side's midpoints should be S/2√3.

S/2√3=√(√3*π)*R/2√3

It's kind of messy, but what I'm getting at is that in order to get 8 equal pieces, you first slice from chord that is √(√3*π)/2√3 times the radius of the cake away from the center to the opposite chord on the opposite side of the cake to make a slice. Make 3 slices just like the one described, 120 degrees away from each other, and you get 8 equal volume pieces.
  Posted by Tristan on 2003-12-11 00:58:19

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