Cannonball problem

Mathematical problem on packing efficiency

A square pyramid of cannonballs in a square frame

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

4900 cannonballs can be arranged as either a square of side 70 or a square pyramid of side 24

Lucas conjectured that the only solutions are (N,M) = (0,0), (1,1), and (24,70), 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][8]

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

See also

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

References

1. Darling, David. "Cannonball Problem". The Internet Encyclopedia of Science.
2. Ma, De Gong (1985). "An Elementary Proof of the Solutions to the Diophantine Equation ${\displaystyle 6y^{2}=x(x+1)(2x+1)}$". Sichuan Daxue Xuebao. 4: 107–116.
3. Anglin, W. S. (1990). "The Square Pyramid Puzzle". American Mathematical Monthly. 97 (2): 120–124. doi:10.2307/2323911. JSTOR 2323911.
4. "week95". Math.ucr.edu. 1996-11-26. Retrieved 2012-01-04.
5. Sloane, N. J. A. (ed.). "Sequence A000292 (Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. Sloane, N. J. A. (ed.). "Sequence A039596 (Numbers that are simultaneously triangular and square pyramidal)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. Weisstein, Eric W. "Square Pyramidal Number". MathWorld.

External links

• Weisstein, Eric W. "Cannonball Problem". MathWorld.