Answer :
Though this problem might strike the novice as
being rather difficult, it is, as a matter of fact, quite easy, and is
made still easier by inserting four out of the ten numbers.
First,
it will be found that squares that are diametrically opposite have a
common difference.
For example, the difference between the square of 14
and the square of 2, in the diagram, is 192; and the difference between
the square of 16 and the square of 8 is also 192.
This must be so in
every case.
Then it should be remembered that the difference between
squares of two consecutive numbers is always twice the smaller number
plus 1, and that the difference between the squares of any two numbers
can always be expressed as the difference of the numbers multiplied by
their sum.
Thus the square of 5 (25) less the square of 4 (16) equals
(2 × 4) + 1, or 9; also,
the square of 7 (49) less the square of 3 (9) equals
(7 + 3) × (7  3),
or 40.
Now, the number 192, referred to above, may be
divided into five different pairs of even factors:
2 × 96, 4 × 48,
6 × 32, 8 × 24,
and 12 × 16, and these divided by 2 give
us, 1 × 48,
2 × 24, 3 × 16,
4 × 12, and 6 × 8.
The difference
and sum respectively of each of these pairs in turn produce 47, 49; 22,
26; 13, 19; 8, 16; and 2, 14.
These are the required numbers, four of
which are already placed.
The six numbers that have to be added may be
placed in just six different ways, one of which is as follows, reading
round the circle clockwise: 16, 2, 49, 22, 19, 8, 14, 47, 26, 13.
I will just draw the reader's attention to one
other little point.
In all circles of this kind, the difference between
diametrically opposite numbers increases by a certain ratio, the first
numbers (with the exception of a circle of 6) being 4 and 6, and the
others formed by doubling the next preceding but one.
Thus, in the
above case, the first difference is 2, and then the numbers increase by
4, 6, 8, and 12. Of course, an infinite number of solutions may be
found if we admit fractions.
The number of squares in a circle of this
kind must, however, be of the form 4n + 6; that is,
it must be a number composed of 6 plus a multiple of 4.
