Pyramid Scheme

Probability Level 3

Consider a pyramid with 2017 rows made up of empty boxes. The first (top) row has 1 box, and each successive row has an additional box so that the 2017th,2017^\text{th}, bottom row has 2017 boxes.

First, place each of the first 2017 positive integers into the boxes in the bottom row. Then, each empty box in the 2016th2016^\text{th} row is filled with the sum of the two numbers beneath it. Then, each successively higher row of boxes is filled in the same way.

Let MM and mm be the maximum and minimum possible values, respectively, of the single box on the top. Then M+m=a×2b,M + m = a \times 2^b, where aa and bb are positive integers such that bb is as large as possible. Find the value of a+b.a + b.

Note: Below is an example of how a pyramid of 4 rows could be filled.


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