Puzzles for Programmers and Pros

1.1. Sweet Tooth

Two children, perhaps similar to ones you know, love cake and mathematics. For this reason, Jeremy convinces Marie to play the following game on two identical rectangular cakes chef Martine has prepared for them.

Jeremy will cut the first cake into two pieces, perhaps evenly, perhaps not. After seeing the cut, Marie will decide whether she will choose first or allow Jeremy to do so. If she goes first, she will take the larger piece. If she goes second, she can assume that Jeremy will take the larger piece.

Next, Jeremy will cut the second cake into two pieces (remember that one of the pieces can be vanishingly small if he so chooses). If Marie had chosen first for the first cake, then Jeremy gets to take the larger piece of the second cake. If Marie had chosen second for the first cake, then she gets to take the larger piece of the second cake.

1.1.1. Warm-Up

Assuming each child will strive to get the most total cake possible, what is an optimal strategy for Jeremy?

NOTE

Before you look at the solution: Assume that Jeremy divides the first cake into fractions f and 1-f where f is at least ½. Then explore the consequences if Marie chooses to take the piece of fraction f or if she goes second, so gets the piece having fraction 1-f.

1.1.2. Solution to Warm-Up

Following the hint, Marie reasons as follows: If she takes the fraction f piece, then Jeremy will take ...