92 THE FASCINATION OF NUMBERS

4. Certain other square numbers have roots which, when
reversed and squared, give the reverse of the original
numbers.

12°=144 and 212=441
13°=169 and 312=q61

5. Some numbers when multiplied give the same digits
but in reverse order.

2178 X4 =8712
1089 X9 =9g8o1
4356 X 13=6534
It will be noted that all these numbers are multiples of 108g.

6. Some numbers, taken together and added, give the
same result as when they are multiplied.

2 X2=4 and 2 +2=4
13X3=4} and 1}43=4}

Apart from the first example given above, one of the num-
bers in each pair must always be in the form of a fraction,
and there is a simple rule which enables such pairs to be
generated.

If x be one of the numbers, then the other will be a frac-
tion of the form ——.

X—1I

Adding these two we get:

 

and multiplying:

X X2
X o
Rt ) »—1

Thus, 74+I=7xI=42

 

 

This relationship is derived from the fact that the square of
any number is equal to unity plus the product of the pre-
ceding and succeeding numbers.

7. The number 123456789 (that is, containing all nine