Cannonball problem

<h2 id="formulation-as-a-diophantine-equation">Formulation as a Diophantine equation</h2>
When cannonballs are stacked within a square frame, the number of balls is a square pyramidal number; <a href="/facts/Thomas_Harriot/rzo8f1Ll">Thomas Harriot</a> gave a formula for this number around 1587, answering a question posed to him by Sir <a href="/facts/Walter_Raleigh/4XLgHJI1">Walter Raleigh</a> on their expedition to America.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> 
<a href="/facts/%C3%89douard_Lucas/NTC4y2N0">Édouard Lucas</a> formulated the cannonball problem as a <a href="/facts/Diophantine_equation/mYaHp7oX">Diophantine equation</a>

∑
          
            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}.}

<h2 id="solution">Solution</h2>

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 <a href="/facts/G._N._Watson/VLfFjgjh">G. N. Watson</a> found a proof for this fact, using <a href="/facts/Elliptic_function/mF03oI04">elliptic functions</a>. More recently, <a href="/facts/Elementary_proof/zv8dBPXx">elementary proofs</a> have been published.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="applications">Applications</h2>
The solution N = 24, M = 70 can be used for constructing the <a href="/facts/Leech_lattice/LVM1TV0f">Leech lattice</a>. The result has relevance to the <a href="/facts/Bosonic_string_theory/ptmp5HnW">bosonic string theory</a> in 26 dimensions.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
Although it is possible to <a href="/facts/Squaring_the_square/8LPIGM4t">tile a geometric square with unequal squares</a>, 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.

<h2 id="related-problems">Related problems</h2>
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.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
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.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
The only numbers that are simultaneously <a href="/facts/Triangular_number/KHonHtQJ">triangular</a> and square pyramidal are 1, 55, 91, and 208335.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
There are no numbers (other than the trivial solution 1) that are both <a href="/facts/Tetrahedral_number/b0hMLL5J">tetrahedral</a> and square pyramidal.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Square_triangular_number/mHt764IS">Square triangular number</a>, the numbers that are simultaneously square and triangular</li>
<li><a href="/facts/Close-packing_of_equal_spheres/AqL1HUxe">Close-packing of equal spheres</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/CannonballProblem.html">"Cannonball Problem"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Darling, David. "Cannonball Problem". The Internet Encyclopedia of Science. <a href="http://www.daviddarling.info/encyclopedia/C/Cannonball_Problem.html" target="_blank">http://www.daviddarling.info/encyclopedia/C/Cannonball_Problem.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Ma, De Gang (1984). "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)}
 
". Chinese Science Bulletin. 29 (21): 1343–1343. doi:10.1360/csb1984-29-21-1343. <a href="https://www.sciengine.com/doi/pdfView/6949ebef4d2c4985a6b6707fb16676c7" target="_blank">https://www.sciengine.com/doi/pdfView/6949ebef4d2c4985a6b6707fb16676c7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Anglin, W. S. (1990). "The Square Pyramid Puzzle". American Mathematical Monthly. 97 (2): 120–124. doi:10.2307/2323911. JSTOR 2323911. <a href="/wiki/American_Mathematical_Monthly" target="_blank">/wiki/American_Mathematical_Monthly</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"week95". Math.ucr.edu. 1996-11-26. Retrieved 2012-01-04. <a href="http://math.ucr.edu/home/baez/week95.html" target="_blank">http://math.ucr.edu/home/baez/week95.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Weisstein, Eric W. "Square Pyramidal Number". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Weisstein, Eric W. "Square Pyramidal Number". MathWorld. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Cannonball problem open-in-new

Cannonball problem