Cannonball problem explained

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

\sum ^{N} _{n =1} n ^{2} = M ^{2} or(1)/(6) N(N +1 )( 2 N +1 ) = (2 N ^{3+3N} ^{2+N})/(6) = M ^{2.}

Solution

Lucas conjectured that the only solutions are,, and, using either 0, 1, or 4900 cannonballs. 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 cannonball problem, which is to yield a perfect square from the N^th 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 cannonballs.^[5]

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

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

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

See also

• Square triangular number, the numbers that are simultaneously square and triangular
• Close-packing of equal spheres

Notes and References

1. Encyclopedia: Cannonball Problem . The Internet Encyclopedia of Science . Darling . David.
2. Ma, De Gang . An Elementary Proof of the Solutions to the Diophantine Equation 6 y ^{2=x(x+1)(2x+1)} . Chinese Science Bulletin . 29 . 21 . 1343 - 1343 . 1984. 10.1360/csb1984-29-21-1343 .
3. Anglin, W. S. . The Square Pyramid Puzzle. 2323911 . . 97 . 2 . 120–124 . 1990 . 10.2307/2323911.
4. Web site: week95 . Math.ucr.edu . 1996-11-26 . 2012-01-04.
5. A000292. Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.
6. A002411. Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2.
7. A039596. Numbers that are simultaneously triangular and square pyramidal.