a
square that shall contain exactly the same area, you are confronted with
the problem of squaring the circle. Well, it cannot be done with
exactitude (though we can get an answer near enough for all practical
purposes), because it is not possible to say in exact numbers what is the
ratio of the diameter to the circumference. But it is only in recent
times that it has been proved to be impossible, for it is one thing not
to be able to perform a certain feat, but quite another to prove that it
cannot be done. Only uninstructed cranks now waste their time in trying
to square the circle.

Again, we can never measure exactly in numbers the diagonal of a square.
If you have a window pane exactly a foot on every side, there is the
distance from corner to corner staring you in the face, yet you can never
say in exact numbers what is the length of that diagonal. The simple
person will at once suggest that we might take our diagonal first, say an
exact foot, and then construct our square. Yes, you can do this, but then
you can never say exactly what is the length of the side. You can have it
which way you like, but you cannot have it both ways.

All my readers know what a magic square is. The numbers 1 to 9 can be
arranged in a square of nine cells, so that all the columns and rows and
each of the diagonals will add up 15. It is quite easy; and there is only
one way of doing it, for we do not count as different the arrangements
obtained by merely turning round the square and reflecting it in a
mirror. Now if we wish to make a magic square of the 16 numbers, 1 to 16,
there are just 880 different ways of doing it, again not counting
reversals and reflections. This has been finally proved of recent years.
But how many magic squares may be formed with the 25 numbers, 1 to 25,
nobody knows, and we shall have to extend our knowledge in certain
directions before we can hope to solve the puzzle. But it is surprising
to find that exactly 174,240 such squares may be formed of one particular
restricted kind only--the bordered square, in which the inner square of
nine cells is itself magic. And I have shown how this number may be at
once doubled by merely converting every bordered square--by a simple
rule--into a non-bordered one.

Then vain attempts have been made to construct a magic square by what is
called a "knight's tour" over the chess-board, numbering each square that
the knight visits in succession, 1, 2, 3, 4, etc.; and it has