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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=1}^{N}n^{2}=M^{2}

or

{\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 $(N, M) = (0,0)$ , $(1,1)$ , 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.

References

^ Darling, David. "Cannonball Problem". The Internet Encyclopedia of Science.
^ Ma, De Gang (1984). "An Elementary Proof of the Solutions to the Diophantine Equation $6y^{2}=x(x+1)(2x+1)$ ". Chinese Science Bulletin. 29 (21): 1343–1343. doi:10.1360/csb1984-29-21-1343.
^ Anglin, W. S. (1990). "The Square Pyramid Puzzle". American Mathematical Monthly. 97 (2): 120–124. doi:10.2307/2323911. JSTOR 2323911.
^ "week95". Math.ucr.edu. 1996-11-26. Retrieved 2012-01-04.
^ 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.
^ 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.
^ Sloane, N. J. A. (ed.). "Sequence A039596 (Numbers that are simultaneously triangular and square pyramidal)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ ^a ^b Weisstein, Eric W. "Square Pyramidal Number". MathWorld.

External links

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

[1] Darling, David. "Cannonball Problem". The Internet Encyclopedia of Science.

[2] Ma, De Gang (1984). "An Elementary Proof of the Solutions to the Diophantine Equation $6y^{2}=x(x+1)(2x+1)$ ". Chinese Science Bulletin. 29 (21): 1343–1343. doi:10.1360/csb1984-29-21-1343.

[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.

[Weisstein-8] Weisstein, Eric W. "Square Pyramidal Number". MathWorld.

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@@ Line 4: / Line 4: @@
 Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1.
-==Formulation as a Diophantine equation==
+==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.<ref>{{cite encyclopedia |title=Cannonball Problem |encyclopedia=The Internet Encyclopedia of Science |author=David Darling |url=http://www.daviddarling.info/encyclopedia/C/Cannonball_Problem.html }}</ref>
+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.<ref>{{cite encyclopedia |title=Cannonball Problem |encyclopedia=The Internet Encyclopedia of Science |url=http://www.daviddarling.info/encyclopedia/C/Cannonball_Problem.html |last=Darling |first=David}}</ref>
 [[Édouard Lucas]] formulated the cannonball problem as a [[Diophantine equation]]
 :<math>\sum_{n=1}^{N} n^2 = M^2</math>
 or
-:<math display="block">\frac{1}{6} N(N+1)(2N+1) = \frac{2N^3+3N^2+N}{6} = M^2.</math>
+:<math>\frac{1}{6} N(N+1)(2N+1) = \frac{2N^3+3N^2+N}{6} = M^2.</math>
 ==Solution==
 [[File:Cannonball_problem.svg|thumb|300px|4900 cannonballs can be arranged as either a square of side 70 {{nowrap|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 function]]s.  More recently, [[elementary proof]]s have been published.<ref>{{Cite journal|author=Ma, De Gong |title=An Elementary Proof of the Solutions to the Diophantine Equation <math>6y^2=x(x+1)(2x+1)</math> |journal=Sichuan Daxue Xuebao |volume=4 |pages=107–116 |year=1985}}</ref><ref>{{Cite journal|author=Anglin, W. S. |title=The Square Pyramid Puzzle|jstor=2323911 |journal=[[American Mathematical Monthly]] |volume=97 |issue=2 |pages=120–124 |year=1990 |doi=10.2307/2323911}}</ref>
+Lucas conjectured that the only solutions are {{math|1=(''N'',''M'') = (0,0)}}, {{math|1=(1,1)}}, and {{math|1=(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 function]]s.  More recently, [[elementary proof]]s have been published.<ref>{{Cite journal|author=Ma, De Gang |title=An Elementary Proof of the Solutions to the Diophantine Equation <math>6y^2=x(x+1)(2x+1)</math> |journal=Chinese Science Bulletin |volume=29 |issue=21 |pages=1343 - 1343 |year=1984|doi=10.1360/csb1984-29-21-1343 |url=https://www.sciengine.com/doi/pdfView/6949ebef4d2c4985a6b6707fb16676c7}}</ref><ref>{{Cite journal|author=Anglin, W. S. |title=The Square Pyramid Puzzle|jstor=2323911 |journal=[[American Mathematical Monthly]] |volume=97 |issue=2 |pages=120–124 |year=1990 |doi=10.2307/2323911}}</ref>
 ==Applications==
@@ Line 21: / Line 21: @@
 ==Related problems==
-A triangular-pyramid version of the Cannon Ball Problem, which is to yield a perfect square from the ''N''<sup>th</sup> Tetrahedral number, would have ''N'' = 48. <!-- N=1, and N=2, are also solutions. --> That means that the (24 × 2 = ) 48th tetrahedral number equals to (70<sup>2</sup> × 2<sup>2</sup> = 140<sup>2</sup> = ) 19600. This is comparable with the 24<sup>th</sup> square pyramid having a total of 70<sup>2</sup> cannon balls.<ref>{{Cite OEIS|sequencenumber=A000292|name=Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6}}</ref>
+A triangular-pyramid version of the cannonball problem, which is to yield a perfect square from the ''N''<sup>th</sup> Tetrahedral number, would have ''N'' = 48. <!-- N=1, and N=2, are also solutions. --> That means that the (24 × 2 = ) 48th tetrahedral number equals to (70<sup>2</sup> × 2<sup>2</sup> = 140<sup>2</sup> = ) 19600. This is comparable with the 24th square pyramid having a total of 70<sup>2</sup> cannonballs.<ref>{{Cite OEIS|sequencenumber=A000292|name=Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6}}</ref>
-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.<ref>{{Cite OEIS|sequencenumber=A002411|name=Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2}}</ref>
+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.<ref>{{Cite OEIS|sequencenumber=A002411|name=Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2}}</ref>
-The only numbers that are simultaneously [[triangular number|triangular]] and square pyramidal, are 1, 55, 91, and 208335.<ref>{{Cite OEIS|sequencenumber=A039596|name=Numbers that are simultaneously triangular and square pyramidal}}</ref><ref name="Weisstein">{{MathWorld|id=SquarePyramidalNumber|title=Square Pyramidal Number}}</ref>
+The only numbers that are simultaneously [[triangular number|triangular]] and square pyramidal are 1, 55, 91, and 208335.<ref>{{Cite OEIS|sequencenumber=A039596|name=Numbers that are simultaneously triangular and square pyramidal}}</ref><ref name="Weisstein">{{MathWorld|id=SquarePyramidalNumber|title=Square Pyramidal Number}}</ref>
 There are no numbers (other than the trivial solution 1) that are both [[tetrahedral number|tetrahedral]] and square pyramidal.<ref name="Weisstein"/>