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Cannonball problem

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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

or

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 = 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.

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]x2=) 48th tetrahedral number equals to ([70²]x2²=140²=) 19600. This is comparable with [24]th square pyramid yielding a total of [70²] cannon balls. (Reference : Tetrahedral Numbers : oeis.org/A000292/ )


Similarly, a pentagonal-pyramid version of the Cannon Ball problem to produce a perfect square, would have N=8, summing a total of (14x14=) 196 cannon balls. (Reference : Pentagonal Numbers : oeis.org/A002411 )

The only numbers that are simultaneously triangular and square pyramidal, are 1, 55, 91, and 208335.[5][6]

There are no numbers (other than the trivial solution 1) that are both tetrahedral and square pyramidal.[6]

See also

References

  1. ^ David Darling. "Cannonball Problem". The Internet Encyclopedia of Science.
  2. ^ Ma, D. G. (1985). "An Elementary Proof of the Solutions to the Diophantine Equation ". 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 A039596 (Numbers that are simultaneously triangular and square pyramidal)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ a b Weisstein, Eric W. "Square Pyramidal Number". MathWorld.
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