# Pile Splitting: Freaky Fractions, Crazy 1s, Matrices, and More!

Here’s a curious puzzle, a classic.

*Take a pile of nine pebbles and split them into piles. Say, piles of 4 and 5. On the side, write 5 x 4 = 20 . Now split the pile of 5 into two, say piles of 2 and 3. Write 2 x 3= 6 . Keep doing this—splitting piles and recording products—until you have nine piles each of 1 pebble.*

Now sum up the products. You are sure to get the magic sum of 36.

(Try it! Make different splitting choices along the way. You cannot escape the magic sum of 36.)

But this is just the beginning:

FREAKY FRACTIONS:

Freaky Fraction Splitting VIDEO

CRAZY 1s:

And here is a PDF explaining the mathematics behind this–and more!–including ideas on how you, yourself, can discover even more freaky arithmetic invariants in pile-splitting activity!