We were required to find the smallest number of
cannon balls that we could lay on the ground to form a perfect square,
and could pile into a square pyramid.
I will try to make the matter clear to the merest
Here in the first row we place in regular order
the natural numbers.
Each number in the second row represents the sum of the numbers in the
row above, from the beginning to the number just over it. Thus 1, 2, 3,
4, added together, make 10.
The third row is formed in exactly the same way as the second. In the
fourth row every number is formed by adding together the number just
above it and the preceding number.
Thus 4 and 10 make 14, 20 and 35 make 55.
Now, all the numbers in the second row are triangular numbers, which
means that these numbers of cannon balls may be laid out on the ground
so as to form equilateral triangles.
The numbers in the third row will all form our triangular pyramids,
while the numbers in the fourth row will all form square pyramids.
Thus the very process of forming the above numbers
shows us that every square pyramid is the sum of two triangular
pyramids, one of which has the same number of balls in the side at the
base, and the other one ball fewer.
If we continue the above table to twenty-four places, we shall reach
the number 4,900 in the fourth row.
As this number is the square of 70, we can lay out the balls in a
square, and can form a square pyramid with them.
This manner of writing out the series until we come to a square number
does not appeal to the mathematical mind, but it serves to show how the
answer to the particular puzzle may be easily arrived at by
As a matter of fact, I confess my failure to discover any number other
than 4,900 that fulfils the conditions, nor have I found any rigid
proof that this is the only answer.
The problem is a difficult one, and the second answer, if it exists
(which I do not believe), certainly runs into big figures.
For the benefit of more advanced mathematicians I
will add that the general expression for square pyramid numbers is (2n3 + 3n2 + n)/6.
For this expression to be also a square number (the special case of 1
excepted) it is necessary that n = p2 - 1 = 6t2,
where 2p2 - 1 = q2
(the "Pellian Equation").
In the case of our solution above, n = 24, p = 5,
t = 2, q = 7.