Question: 4. Proof by Induction The Stacking Game. You begin with a stack of n boxes. Then you make a sequence of moves. In each move,

4. Proof by Induction The Stacking Game. You begin with a stack of n boxes. Then you make a sequence of moves. In each move, you divide one stack of boxes into two nonempty stacks. The game ends when you have n stacks, each containing a single box. You earn points for each move; in particular, if you divide one stack of height a+b into two stacks with heights a and b, then you score ab points for that move. Your overall score is the sum of the points that you earn for each move. An example of this game for n 8 is shown below. On each line, the underlined stack is divided in the next step. Is there a strategy that you can use to maximize your total score? Prove the claim on the next page to answer this question. [3 Marks] Stack Heights Score in Points 15 3 2 3 3 2 2 1 3 211 1 2 1 21 1 1 28 Total Score

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