For the square cake, the answer involves cutting wedges to the center of the cake. The area of each triangle is given as one half times the base times the height. They all have the same height (one half the length of the side), so as long as you make the bases equal, the areas will be equal too. But having the bases equal also impllies that the frosting along the sides of the cake will be distributed equally. So both the cake and the frosting are shared fairly.
(**Diagrams are not to scale - just rough sketches to illustrate our descriptions)
The idea changes a little for the rectangular cake. The answer becomes harder. You cut a line along the center, but not extending all the way to the edges. Some pieces are cut from the left or right side to the endpoint of the line (forming triangles), and some are cut from the front or back to include pieces of the line (forming trapezoids).
In the 6x12 cake, our center line starts 4 inches from the left-hand edge of the cake and ends 4 inches from the right-hand edge. In general, in an HxL cake, the line starts at distance HL/(H+L) from the left-hand side, and ends the same distance from the right-hand side, so that it has length L(L-H)/(H+L).
We have two types of pieces: triangular pieces, with base on the left-hand side of the cake and apex at the left endpoint of the center line (or base on the right-hand side and apex at the right endpoint of the center line), and trapezoidal pieces, with one base on the bottom or top of the cake and the other base taking a portion of the center line, the length of that portion being proportional to the length of the base.
Then allocate 1/N of the perimeter of the cake to each person. (One person might get part of a triangle and part of a trapezoid, but that's okay.) We only need to show that the ratio of (area of the cake piece) to (length along the perimeter of the whole cake) is a constant, because then each person will get 1/N of the area (thus 1/N of the cake and 1/N of the top frosting) as well as 1/N of the perimeter (and thus 1/N of the side frosting). One can see that this ratio is constant for all the triangles, and constant for all the trapezoids, and the choice of distances ensures that the two ratios are equal.
To see this, the whole cake has area LH and perimeter 2(L+H), So its area:perimeter ratio is LH/(2(L+H)). The left triangle has area (1/2)(H)(LH/(L+H)) and occupies H of the perimeter of the big cake, so that is ratio is also LH/(2(L+H)).
A second solution was provided by ZhenChao Wu, a Grad Student in Computer Science from Northeastern University. Start by drawing a diagonal from the upper-left corner to the lower-right corner. Draw a 45-degree line from the lower-left corner, which intersects the diagonal in a point A,and another 45-degree line from the upper-right corner, which intersects the diagonal at B. The point A is equidistant from the left side and the bottom (in fact distance LH/(L+H)), and the point B is the same distance from the right side and bottom. Now cut triangles, with bases on the left side or bottom and apex at A, or with bases on the right side or top and apex at B. Allocate 1/N of the perimeter to each person, and argue as before that the ratios area:perimeter are constant, so that each person also gets 1/N of the area. The nice thing is that the whole argument proceeds without any calculation this time.
Click here to view his excellent diagram.
Many of you wanted to start by taking all of the frosting off the cake. Another suggestion was to cut the cake into 4N pieces: N triangles to the center from the left side, another N from the bottom, and so on, and gave each diner four pieces, one from the left, one from the bottom, one from the right, and one from the top. Yet another solution involved a food processor set to "liquefy". Very creative - we prefer to call it "thinking outside the box" rather than "cheating".
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