Your task is to, with an input number p, find the smallest positive cannonball number of order p that is NOT 1.


A cannonball number (of order p) is a number which is both:

  • An p-gonal number (See this page).
  • and an p-gonal pyramid number.

    • The nth p-gonal pyramid number is the sum of the 1st to nth p-gonal numbers.
      • (e.g. 4th square pyramid number = 1 + 4 + 9 + 16 = 30)
    • The picture below represents the 4th square pyramid number, as a square pyramid. enter image description here

    • For more info, visit this link.

The cannonball number of order 3, for example, is 10, because it is:

  • The fourth triangle number (1 + 2 + 3 + 4 = 10)
  • and the third triangular pyramid number. (1 + 3 + 6 = 10)


NOTE: If you can find (or make) more useful formulae than my ones here, please post it here (or message me on the question chat thing).

  • If you're interested, the formula for the nth p-gonal number is:

enter image description here

  • And the nth p-gonal pyramid number is:

enter image description here


  • p is guaranteed to be larger than 2.
  • The program must check values for a solution for p up to (and including) 2^16. Your program may do anything if no solutions are found for p.
  • Only positive indices for n.

Test cases

  • 3 outputs 10 (4th triangle number, 3rd triangle pyramid number)
  • 4 outputs 4900 (70th square number, 24th square pyramid number)

This is code-golf, so shortest answer in bytes wins.

Note: If you do post a solution, please include a description of how the code works.

Should I start a bounty for a solution which is better and doesn't use my formulae?

  • 2
    \$\begingroup\$ Could you add some more detail about what a cannonball number is? It's not entirely clear from your description. \$\endgroup\$
    – DJMcMayhem
    Commented Jul 6, 2016 at 16:45
  • 3
    \$\begingroup\$ Is it guaranteed that there is an answer for any n? If not, what is the range of n you'll be using? \$\endgroup\$
    – Geobits
    Commented Jul 6, 2016 at 16:47
  • \$\begingroup\$ @DrGreenEggsandIronMan Edited to make a definition. n-gonal and n-gonal pyramid numbers shouldn't need defining. \$\endgroup\$
    – clismique
    Commented Jul 6, 2016 at 16:51
  • \$\begingroup\$ @Geobits Edited to make a range instead of using memory. \$\endgroup\$
    – clismique
    Commented Jul 6, 2016 at 16:51
  • 1
    \$\begingroup\$ @DerpfacePython TL;DR: Cannonball numbers quite rare. Look for yourself. \$\endgroup\$
    – Riker
    Commented Jul 6, 2016 at 16:57

4 Answers 4


Python 3, 129 127 bytes

def f(p):
 while 1:
  for i in range(x*x):
   if i//x*((p-2)*i//x+4-p)/2==i%x*(i%x+1)*((p-2)*i%x+5-p)/6==x:return x

A function that takes input via argument and returns the output.

This is an extremely naïve brute force, and takes a very long time for even moderately large p; the execution time will be ridiculous for anything approaching the given maximum for p of 2^16, but there is no reason why the program would not work, given sufficient time.

There are probably far shorter and faster ways of doing this, but I thought it would be good to post something to get this started off.

How it works

The return value x is initialised to 2, and then the program simply loops over all the p-gonal and p-gonal pyramidal numbers up to order x. If the current p-gonal and p-gonal pyramidal numbers, calculated using the formulas, are equal to each other and to x, then x must be the relevant cannonball number and this is returned. Else, x is incremented, and the program tries again for the new value of x.

In terms of golfing, a Cartesian product is used to collapse the two for-loops for the p-gonal and p-gonal pyramidal numbers into a single loop, and the formulas were factorised further to save a few bytes.

  • \$\begingroup\$ Ummm... this might be a bit too late to say... but... my formula was wrong. You just need to change the polygonal formula. \$\endgroup\$
    – clismique
    Commented Jul 10, 2016 at 16:05
  • \$\begingroup\$ @DerpfacePython Thanks for catching that. \$\endgroup\$ Commented Jul 10, 2016 at 17:50

JavaScript, 111 98 bytes

f=n=>{for(b=c=g=1;b++;)for(p=b*(b*3+b*b*(n-2)-n+5)/6;g<p;c++)if(p==(g=c*(c*n-c-c-n+4)/2))return p}


for(b=2,c=g=1;b<1e6;b++)    // run index b from 2 (to 100k)
        p=(b*b*3+b*b*b*(n-2)-b*(n-5))/6 // p=the b-th n-pyramidal number
        ;g<p&&c<1e6;c++)   // run while n-gonal is lower than n-pyramidal (and c below 100k)
            g=(c*c*(n-2)-c*(n-4))/2     // g=the c-th n-gonal number
        )) return p                     // if they are equal, return

c is not reinitialized in the inner loop because the next p[b] is definitely larger than the current g[c] (so we have to move on anyway)


for(i in samples) document.write('n='+(n=samples[i])+': '+f(n)+'<br>');

C, 107 bytes

#define X(x)(1+p/2.0*x)*++x
c(p,m,n,a,b){m=n=a=b=1;p-=2;do if(a>b)b+=X(n);else a=X(m);while(a^b);return a;}

Ungolfed with test parameters:

#include <stdio.h>

#define X(x)(1+p/2.0*x)*++x
int c(int p)
    int m = 1, n = 1, a = 1, b = 1;
    p -= 2;
        if(b < X(m))
            b += X(n);
            a = X(m);
    while(a != b);
    return a;

int main()
    printf("%d\n", c(3));
    printf("%d\n", c(4));

This uses the fact that the n-th p-gonal number can be defined as n(1+(p-2)(n-1)/2) and the pyramid number is the sum of the aforementioned numbers.

I think it can be further golfed, given that it's not really necessary for variable a to be saved.

  • \$\begingroup\$ Wait... is the i in your formula the imaginary number i? \$\endgroup\$
    – clismique
    Commented Jul 13, 2016 at 16:57
  • \$\begingroup\$ Oops, sorry. i is supposed to be n. I had different notation littered in my research. I can't imagine using an imaginary number for this problem, and I definitely can't imagine using it in C. \$\endgroup\$
    – user55852
    Commented Jul 13, 2016 at 17:09

old PHP program, 115 106 bytes

+16 for current PHP, see below

  • loops forever
  • usage: with PHP<4.2: run from browser with <scriptpath>?n=<number>
    with PHP<5.4, add register_globals=1 to php.ini (+18 ?)
  • see my JavaScript answer for description
  • +10 for PHP>=5.4: replace 1 with $n=$_GET[n]. Or replace 1 with $n=$argv[1], run php -f <filename> <number>.
  • +6 for finite loop on success: replace echo$p with die(print$p)
  • +/- 0 for function:

    function f($n){for($b=2;1;$b++)for($p=$b*($b*3+$b*$b*($n-2)-$n+5)/6;$g<$p;++$c)if($p==$g=$c*($c*$n-2*$c-$n+4)/2)return$p;}

    loops forever if it finds nothing. Replace 1 with $p<1e6 to break at 100k or with $p<$p+1 to loop until integer overflow. (tested with PHP 5.6)

examples (on function)

foreach($samples as $n)echo "<br>n=$n: ", f($n);

examples output

n=3: 10
n=4: 4900
n=6: 946
n=8: 1045
n=10: 175
n=11: 23725
n=14: 441
n=17: 975061
n=23: 10680265
n=26: 27453385
n=30: 23001
n=32: 132361021
n=35: 258815701
n=41: 55202400
n=43: 245905
n=50: 314755
n=88: 48280

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