TITLE: Reference for puzzle on dividing piles and scoring products
QUESTION [8 upvotes]: There is a pile of $n$ items. Every time you divide a pile into two piles, you get a score being the product of the number of items in the two piles. Show that the sum of your scores at the end is always $\binom{n}{2}$.

My question: What are some (preferably well-known) books/articles that discuss or at least mention this puzzle? Is there a name to the puzzle? I guess it's quite famous, and as I want to mention it in an article, I would like to cite it properly instead of just saying that it is "a famous puzzle". Searching Google for "pile product score" doesn't yield useful results.

REPLY [2 votes]: A name that has been published for this puzzle is pile splitting.  Bill Marion wrote about it as an example of strong induction in a discrete mathematics pedagogy collection I edited.  He presented three other versions, given below, and referenced a James Tanton article with many more.  Also, he mentioned that the puzzle is included in Rosen's textbook.

Bill Marion, Pile Splitting Problem: Introducing Strong Induction, in
Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles, MAA Notes #74, ed. Hopkins,
Mathematical Association of America, 2009, 7-10.
James Tanton, A Dozen Questions About: Pile Splitting, Math
Horizons 12 (2004) 28-31.

Here are the other versions from Bill's article.  Write $n = r+s$ for the split (and require $r, s \ne 0$).  See also Allen Knutson's comment.

Take the sum of all scores $rs(r+s)$.
Take the product of all scores $\frac{1}{r} + \frac{1}{s}$.
Take the product of all scores $\binom{r+s}{r}$.