MATHEMATICAL RECREATIONS 179

the right, or the a’th column from the right and 4’th row from
the bottom.

Now the square that is in the @’th row from the top and
the b’th column from the left is symmetrical with regard to
the line of 10’s to the square in the a’th column from the right
and b’th row from the bottom, which verifies what we said,
since @ 4 b and ¢ 4 d are in these respectiv squares.

Our thesis could be proved also from the fact that
(@ + b) + (¢ + d) = 20 or from the fact that the numbers in
the lines parallel to the diagonal line diminish regularly by
unity going to one side of this line and increase by unity
going to the other side.

Either method also shows that the sum of two numbers on
opposit sides of the diagonal and equidistant from it is 20.

462. The fact stated in § 453 helps similarly to explain why
in the multiplication table up to 9 X 9, given below with the
diagonal line of products from the lower left hand corner to
the upper right hand corner omitted, two numbers in squares
symmetrically situated with respect to this diagonal line have
the same units figure.

1]2‘3[4)5‘6'7 8]9