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.
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Submitted by DJ
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Rating: 3.6667 (9 votes)
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Solution:
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Yes, it is possible
However, it is not possible to divide a circle into seven equal sections with three lines.
Three lines that divide a circle into 7 regions create one triangle in the middle, with three radial 'pie pieces,' and three shapes that look like a pie slice with the tip cut off (they share the triangle in the center). Probably the best way to get an idea of that is just to sketch a circle on a piece of paper and draw any three intersecting (but non-colinear) lines.
Assume that all seven regions can be of equal area.
Then each of the three lines, when looked at individually, must divide the circle into two pieces of 3/7 and 4/7 of the circle, since each line has three pieces to one side anf four to the other. These three lines must then be equidistant from the center of the circle.
Then, the lines must be equally spaced around the circle, so that regions of roughly the same shape will be of the same area. This makes a three-way symmetry.
Then, by playing around with these lines, I find that it cannot be done.
If I make the center area about the same area as the three mini pie slices, then the three 'half-slices' are much too large. If I increase the size of the mini slices, then the center area gets even smaller.
This experimentation is enough to qualify as a good informal proof that it cannot be done.
Dividing a circle into six equal regions with three straight cuts is fairly trivial and can be accomplished in several different ways. The most obvious is to make three diametric cuts at 60° from each other; this is the 'normal' way to divide a circle into sixths anyway. A diameter with the the other two lines intersecting perpendicularly at appropriate distances would not be too hard to figure out, and other more creative solutions with angled lines are possible as well.
However, I am not dividing a circle; I am cutting a cake. So, I can make the first two cuts into four equal parts, each of which is a quarter-circular slice of cake. Then, without moving the pieces, I make a third straight cut, horizontally, through the middle of the height of the cake, cutting each cake into two pieces of equal area. The slices now, then, are eight quarter-circles, each of which has half the height of the original cake.
Of course, only four of these pieces now have frosting on top (unless it was a double-decker cake to begin with), so there may still be some quibble about that... |
