## THE SULTAN'S ARMY.

(

Money Puzzles)

A certain Sultan wished to send into battle an army that could be formed

into two perfect squares in twelve different ways. What is the smallest

number of men of which that army could be composed? To make it clear to

the novice, I will explain that if there were 130 men, they could be

formed into two squares in only two different ways--81 and 49, or 121

and 9. Of course, all the men must be used on every occasion.

## Answer:

The smallest primes of the form 4n + 1 are 5, 13, 17, 29, and 37, and

the smallest of the form 4n - 1 are 3, 7, 11, 19, and 23. Now, primes of

the first form can always be expressed as the sum of two squares, and in

only one way. Thus, 5 = 4 + 1; 13 = 9 + 4; 17 = 16 + 1; 29 = 25 + 4; 37

= 36 + 1. But primes of the second form can never be expressed as the

sum of two squares in any way whatever.

In order that a number may be expressed as the sum of two squares in

several different ways, it is necessary that it shall be a composite

number containing a certain number of primes of our first form. Thus, 5

or 13 alone can only be so expressed in one way; but 65, (5 x 13), can

be expressed in two ways, 1,105, (5 x 13 x 17), in four ways, 32,045, (5

x 13 x 17 x 29), in eight ways. We thus get double as many ways for

every new factor of this form that we introduce. Note, however, that I

say _new_ factor, for the _repetition_ of factors is subject to another

law. We cannot express 25, (5 x 5), in two ways, but only in one; yet

125, (5 x 5 x 5), can be given in two ways, and so can 625, (5 x 5 x 5 x

5); while if we take in yet another 5 we can express the number as the

sum of two squares in three different ways.

If a prime of the second form gets into your composite number, then that

number cannot be the sum of two squares. Thus 15, (3 x 5), will not

work, nor will 135, (3 x 3 x 3 x 5); but if we take in an even number of

3's it will work, because these 3's will themselves form a square

number, but you will only get one solution. Thus, 45, (3 x 3 x 5, or 9 x

5) = 36 + 9. Similarly, the factor 2 may always occur, or any power of

2, such as 4, 8, 16, 32; but its introduction or omission will never

affect the number of your solutions, except in such a case as 50, where

it doubles a square and therefore gives you the two answers, 49 + 1 and

25 + 25.

Now, directly a number is decomposed into its prime factors, it is

possible to tell at a glance whether or not it can be split into two

squares; and if it can be, the process of discovery in how many ways is

so simple that it can be done in the head without any effort. The number

I gave was 130. I at once saw that this was 2 x 5 x 13, and consequently

that, as 65 can be expressed in two ways (64 + 1 and 49 + 16), 130 can

also be expressed in two ways, the factor 2 not affecting the question.

The smallest number that can be expressed as the sum of two squares in

twelve different ways is 160,225, and this is therefore the smallest

army that would answer the Sultan's purpose. The number is composed of

the factors 5 x 5 x 13 x 17 x 29, each of which is of the required form.

If they were all different factors, there would be sixteen ways; but as

one of the factors is repeated, there are just twelve ways. Here are the

sides of the twelve pairs of squares: (400 and 15), (399 and 32), (393

and 76), (392 and 81), (384 and 113), (375 and 140), (360 and 175), (356

and 183), (337 and 216), (329 and 228), (311 and 252), (265 and 300).

Square the two numbers in each pair, add them together, and their sum

will in every case be 160,225.