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The cannonballs that make a square and a pyramid

numberspyramidal-numberssquaresclassic

Cannonballs are stacked in a square pyramid: one ball on top, a 2 by 2 layer beneath it, then 3 by 3, and so on down to an n by n base. A sergeant notices that the same balls, laid out flat, would form a perfect square.

  1. How many cannonballs are there? (More than one, that is; a single ball trivially works.)
  2. Write down the formula for the number of balls in a pyramid with n layers and show how you would search.
Show a hint

The pyramid with n layers holds 1 + 4 + 9 + ... + n squared = n(n + 1)(2n + 1)/6 balls. Try n up to about 30.

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