The pile-splitting problem
M117, October 12, 2011 (due October 14, 2011)
Your names:
1. The Pile Problem
Given n objects in a pile, split the objects into two smaller piles. Continue to split each
pile into two smaller piles until there are n piles of size one. At each splitting, compute the
product of the size of the two smaller piles. Once there are n piles, sum all the products
computed. The result will always be the same no matter how each of the piles is split into
two smaller piles. The sum of the products is a function of n. We will make a conjecture for
this sum and prove that it is correct.
(1) Have one person calculate the sum described above for n = 8 by taking a pile with 8
objects and splitting them up, computing the products of the split piles each time,
then adding all of these products together. What number do you get?
(2) Have the other person do the same thing, but this time splitting up the 8 objects in
a different way. Did you get the same number as in question (1)?
(3) Now try this for smaller values of n and fill in the values you get in the table below.
(The first few values are filled in for you.)
n sum of split products
2
1
3
3
4
6
5
6
7
8
(4) Conjecture a formula (in terms of n) for the sum of the split products.
(5) Prove the formula you found in (4) by mathematical induction.