# cannonball problem

A tetrahedral stack of cannonballs.

The mathematical analysis of stacks of cannonballs – and of spheres in general – has its roots in a question posed by Sir Walter Raleigh,
favorite of Queen Elizabeth I, explorer, introducer of the potato and tobacco
to Britain, and part-time pirate on the high seas. Raleigh asked his mathematical
assistant, Thomas Harriot, how he could
quickly figure out the number of cannonballs in a square pyramidal stack
(the accompanying photo is actually of a tetrahedral stack). In other words,
given how many cannonballs ran along the side of the bottom layer, how could
the total number of cannonballs in the stack be figured out without having
to count them individually. Harriot solved this problem without difficulty.
If *k* is the number of cannonballs along the side of the bottom layer,
the number of cannonballs in the pyramid, *n*, is equal to 1/6 *k*(1 + *k*)(1 + 2*k*). For example,
if *k* = 7, *n* = 140.

A more specific form of the cannonball problem asks what is the smallest
number of balls that can first be laid out on the ground as an *n* × *n* square, then piled into a square pyramid *k* balls
high? In other words, what is the smallest square
number that is also square pyramidal number? This answer is the smallest
solution to the Diophantine equation

1/6 *k*(1 + *k*)(1 + 2*k*) = *n *^{2}

and turns out to be *k* = 24, *n* = 70, corresponding to 4,900
cannonballs. The ultimate form of the cannonball problem is to ask if there
are any other, larger solutions. In 1875, Edouard Lucas conjectured that there weren't and in 1918 G. N. Watson proved that Lucas
was right.^{1}

Returning to Elizabethan times, Thomas Harriot's interest in spheres extended
far beyond piles of cannonballs. Harriot was an atomist (see atomism),
in the classical Greek sense, and believed that understanding how spheres
pack together was crucial to understanding how the basic constituents of
nature are arranged. Harriot also carried out numerous experiments in optics
and was far ahead of his time in this field. So, when, in 1609, Johannes Kepler wanted some advice on how to give
his own theories on optics a stronger scientific underpinning who better
to turn to than the Englishman? Harriot supplied Kepler with important data
on the behavior of light rays passing through glass, but he also stimulated
the German's interest in the sphere-packing problem. In response, Kepler published a little booklet titled *The Six-Cornered
Snowflake* (1611) that would influence the science of crystallography
for the next two centuries and that contained what has come to be known
as Kepler's conjecture about
the most efficient way to pack spheres.

### Reference

1. Watson, G. N. "The Problem of the Square Pyramid." *Messenger
of Mathematics*, 48: 1-22 (1918-1919).