# 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:

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!

Pile Splitting

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QUADRATICS, Permutations and Combinations, EXPLODING DOTS, and more!

Written for educators - and their students too! - this website, slowly growing, takes all the content Tanton has developed in his books, videos, and workshops, and organizes it into short, self-contained, and complete, curriculum units proving that mathematics, at all points of the school curriculum, can be joyous, fresh, innovative, rich, deep-thinking, and devoid of any rote doing! Let's teach generations of students to be self-reliant thinkers, willing to flail and to use their common sense to "nut their way" through challenges, to assess and judge results, and to adjust actions to find success. (Great life skills!)

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