Splitting Coins

You split 1000 coins into two piles and count the number of coins in each pile. If there are X coins in pile one and Y coins in pile two, you multiple the two numbers to get XY. Then you split both piles further, repeating the same counting and multiplication process, and adding the new multiplication results to the first one. The process ends when you end up with 1000 single-coin piles. Prove that you will always get the same final result, no matter how the piles have been divided during the splitting process.

For example, if you start with 5 coins and split them into a pile of 2 and a pile of 3, you get the number 2×3=6. Then, if you split the pile of 3 into a pile of 1 and a pile of 2, you will add 1×2=2 more to the 6 and get 8. Finally, if you split the two piles of 2 into single-coin piles, you will end up with 8+1+1=10.

SOLUTION

Consider the sum of the squares of the numbers of coins in each pile, plus twice the sum of the products. On each step, if you split a pile of X+Y coins into a pile of X coins and a pile of Y coins, the sum of the squares will get reduced by 2XY, exactly the amount the sum of the products will increase by. Therefore, that number remains constant throughout the entire process and ends up exactly (1000²-1000)/2=499500.

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