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

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

or

1 6 N ( N + 1 ) ( 2 N + 1 ) = 2 N 3 + 3 N 2 + N 6 = M 2 . {\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 = 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]

See also

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 6 y 2 = x ( x + 1 ) ( 2 x + 1 ) {\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. ^ a b Weisstein, Eric W. "Square Pyramidal Number". MathWorld.

External links