In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1.

Formulation as a Diophantine equation

When cannonballs are stacked within a square frame, the number of balls is a square pyramidal number; Thomas Harriot gave a formula for this number around 1587, answering a question posed to him by Sir Walter Raleigh on their expedition to America.^[1] Édouard Lucas formulated the cannonball problem as a Diophantine equation

${\displaystyle \sum _{n=1}^{N}n^{2}=M^{2))$

or

$${\displaystyle {\frac {1}{6))N(N+1)(2N+1)={\frac {2N^{3}+3N^{2}+N}{6))=M^{2}.}$$

Solution

Lucas conjectured that the only solutions are N = 1, M = 1, and N = 24, M = 70, using either 1 or 4900 cannon balls. It was not until 1918 that G. N. Watson found a proof for this fact, using elliptic functions. More recently, elementary proofs have been published.^[2][3]

Applications

The solution N = 24, M = 70 can be used for constructing the Leech lattice. The result has relevance to the bosonic string theory in 26 dimensions.^[4]

Although it is possible to tile a geometric square with unequal squares, it is not possible to do so with a solution to the cannonball problem. The squares with side lengths from 1 to 24 have areas equal to the square with side length 70, but they cannot be arranged to tile it.

Related problems

A triangular-pyramid version of the Cannon Ball Problem, which is to yield a perfect square from the Nth Tetrahedral number, would have N = 48. That means that the (24 × 2 = ) 48th tetrahedral number equals to (70² × 2² = 140² = ) 19600. This is comparable with the 24th square pyramid having a total of 70² cannon balls.^[5]

Similarly, a pentagonal-pyramid version of the Cannon Ball problem to produce a perfect square, would have N = 8, yielding a total of (14 × 14 = ) 196 cannon balls.^[6]

The only numbers that are simultaneously triangular and square pyramidal, are 1, 55, 91, and 208335.^[7][8]

There are no numbers (other than the trivial solution 1) that are both tetrahedral and square pyramidal.^[8]