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Grade Vs Difficulty:

This is a lesson plan by James Tanton based on the Pile Splitting activity. It will be presented at the 2011 Circle on the Road workshop.
DESCRIPTION: Start with a collection of nine coins and split them into two piles in any way you like. Note the size of the two piles you created and write down on a piece of paper the product of those two sizes. (So, for example, if you split the coins into a pile of 6 and a pile of 3, write down the number 6x3 = 18.) Now split each of the two piles into smaller piles and record the product of their sizes as well. Keep doing this until you have nine piles of one coin each. Add up all the products you recorded. This is your magic number. Regroup the nine coins and repeat this activity again, making different choices for the splits along the way. Do you see why your number is magic?
The attached lesson plan gives many more details. Discussion and more related files may be found at: https://www.mathcircles.org/content/pilesplitting.
pile splitting problem
I am very excited to work on this. I looked at the problem last weekend before you posted
your notes. I solved it in the most barbaric of ways (a real mathematician's solution!), namely
observing what appears to be the answer and then proving by mathematical induction that it really does always happen actually this is kind of fun and I fouled up the algebra the first time through. then of course once you see what the answer is you can give a combinatorial argument which is much more illuminating. Anyhow, I love the problem and look forward to reading your notes to see where you take it and what other insights you have into this fun problem! I will write again once I have a chance to digest your notes.