Cannonball problem: Difference between revisions
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{{Short description|Mathematical problem on packing efficiency}} |
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[[File:Rye Castle, Rye, East Sussex, England-6April2011 (1).jpg|thumb|A square pyramid of cannonballs in a square frame]] |
[[File:Rye Castle, Rye, East Sussex, England-6April2011 (1).jpg|thumb|A square pyramid of cannonballs in a square frame]] |
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In the mathematics of [[figurate number]]s, the '''cannonball problem''' asks which numbers are both [[square]] and [[square pyramidal number|square pyramidal]]. |
In the mathematics of [[figurate number]]s, the '''cannonball problem''' asks which numbers are both [[Square number|square]] and [[square pyramidal number|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. |
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Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1 |
Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1. |
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==Formulation as a Diophantine equation == |
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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 |
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> |
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[[Édouard Lucas]] formulated the cannonball problem as a [[Diophantine equation]] |
[[Édouard Lucas]] formulated the cannonball problem as a [[Diophantine equation]] |
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:<math>\sum_{n=1}^{N} n^2 = M^2</math> |
:<math>\sum_{n=1}^{N} n^2 = M^2</math> |
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or |
or |
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:<math>\frac{1}{6} N(N+1)(2N+1) = M^2</math> |
:<math>\frac{1}{6} N(N+1)(2N+1) = \frac{2N^3+3N^2+N}{6} = M^2.</math> |
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==Solution== |
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[[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}}]] |
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| ⚫ | 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> |
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==Applications== |
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The solution ''N'' = 24, ''M'' = 70 can be used for constructing the [[Leech lattice#Using the Lorentzian lattice II25,1|Leech lattice]]. The result has relevance to the [[bosonic string theory]] in 26 dimensions.<ref>{{cite web|url=http://math.ucr.edu/home/baez/week95.html |title=week95 |publisher=Math.ucr.edu |date=1996-11-26 |accessdate=2012-01-04}}</ref> |
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Although it is possible to [[Squaring the square|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. |
Although it is possible to [[Squaring the square|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. |
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==Related problems== |
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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> |
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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> |
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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> |
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There are no numbers (other than the trivial solution 1) that are both [[tetrahedral number|tetrahedral]] and square pyramidal.<ref name="Weisstein"/> |
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==See also== |
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*[[Square triangular number]], the numbers that are simultaneously square and triangular |
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*[[Close-packing of equal spheres]] |
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==References== |
==References== |
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{{reflist}} |
{{reflist|2}} |
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==External links== |
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*{{mathworld|id=CannonballProblem|title=Cannonball Problem}} |
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[[Category:Diophantine equations]] |
[[Category:Diophantine equations]] |
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[[Category:Figurate numbers]] |
[[Category:Figurate numbers]] |
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{{math-stub}} |
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Latest revision as of 18:06, 20 October 2024
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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
[edit]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
[edit]
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
[edit]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
[edit]A triangular-pyramid version of the cannonball 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 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
[edit]- Square triangular number, the numbers that are simultaneously square and triangular
- Close-packing of equal spheres
References
[edit]- ^ Darling, David. "Cannonball Problem". The Internet Encyclopedia of Science.
- ^ Ma, De Gang (1984). "An Elementary Proof of the Solutions to the Diophantine Equation ". 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.
