INTRODUCTION
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
squaYes 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 been done with the exception of the two diagonals, which so
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