2. We are playing a mound-splitting game - we start with a single mound of rocks. In each move, we pick a mound, split it into 2 mounds of arbitrary size (say,
rocks), multiply the # of rocks in the 2 mounds and write down the result (
), and continue playing until only one rock remains in each mound. At the end, we add up all the numbers written down after the splits. Prove that if we start with
rocks, then the final sum will be
, no matter how we split the mounds or in which order we split them.Thanks!