Cutting a Cake with Icing
One day, grandma baked a cake with a square top and dimensions 30cm x 30cm x 10cm. What is a simple strategy for cutting the cake into 9 equal pieces? The next day, grandma baked another cake with the same dimensions. This time, she put a thin layer of icing on top and on all four sides but not on the bottom. What is a simple strategy for cutting such a cake into 9 pieces such that all pieces have the same amount of cake by volume and the same amount of icing by surface area?
Let the dimensions of the cake with a square top be s x s x h. Let p denote the number of pieces. The problem can be solved for any values of s, h and p as follows:
Cake without icing: Pieces of size (s/p) x s x h may be carved out with p-1 cuts.
Cake with icing: Identify p points on the edges of the square top that divide its perimeter into p equal lengths. For each of these p points, slice the cake vertically (from top to bottom using a knife) along the line segment joining the center of the square top to these points. To convince yourself that the resulting pieces have equal amounts of cake (by volume) and equal amounts of icing (by surface area), please see the figure attached for p = 7. Draw additional line segments from the vertices of the square top to its center. Now the top of each piece is either a triangle or the union of two triangles. Since the area of a triangle is half the product of its base and height, it is easy to see why the p pieces divide the square equally by area.
Shankar chooses a number between 1 and 10,000. Geeta has to guess the chosen number as quickly as possible. Shankar will let Geeta know whether her guess is smaller than, larger than or equal to the number. The caveat is that Geeta loses the game if her guess is larger than Shankar's chosen number two or more times. (A) How many guesses are necessary? (B) What if Shankar is allowed to pick an arbitrarily large positive number?
Shankar chooses a number uniformly at random between 1 and 1000. Geeta has to guess the chosen number as quickly as possible. Shankar will let Geeta know whether her guess is smaller than, larger than or equal to the number. If Geeta's guess is larger than the number, Shankar replaces the number with another number chosen uniformly at random [1, 1000]. What should Geeta's strategy be?
25 Jan 2011
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