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 anybody.  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 (2n cubed + 3n squared + n)/6.  For this expression to be also a square number (the special case of 1 excepted) it is necessary that n = p squared — 1 = 6t squared, where 2p squared — 1 = q squared (the “Pellian Equation").  In the case of our solution above, n = 24, p = 5, t = 2, q = 7.

139.—­THE DUTCHMEN’S WIVES.

The money paid in every case was a square number of shillings, because they bought 1 at 1s., 2 at 2s., 3 at 3s., and so on.  But every husband pays altogether 63s. more than his wife, so we have to find in how many ways 63 may be the difference between two square numbers.  These are the three only possible ways:  the square of 8 less the square of 1, the square of 12 less the square of 9, and the square of 32 less the square of 31.  Here 1, 9, and 31 represent the number of pigs bought and the number of shillings per pig paid by each woman, and 8, 12, and 32 the same in the case of their respective husbands.  From the further information given as to their purchases, we can now pair them off as follows:  Cornelius and Gurtruen bought 8 and 1; Elas and Katruen bought 12 and 9; Hendrick and Anna bought 32 and 31.  And these pairs represent correctly the three married couples.

The reader may here desire to know how we may determine the maximum number of ways in which a number may be expressed as the difference between two squares, and how we are to find the actual squares.  Any integer except 1, 4, and twice any odd number, may be expressed as the difference of two integral squares in as many ways as it can be split up into pairs of factors, counting 1 as a factor.  Suppose the number to be 5,940.  The factors are 2 squared.3 cubed.5.11.  Here the exponents are 2, 3, 1, 1.  Always deduct 1 from the exponents of 2 and add 1 to all the other exponents; then we get 1, 4, 2, 2, and half the product of these four numbers will be the required number