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THE FASCINATION OF’NUMBERS

here as x. The original number is then known to be
twice the final result (that is, 2x) unless at the second
stage it was necessary to add % in which case the
original number is equivalent to 2x+ 1. The proof is
as follows.

(i) If the original number is even, it is of the form 2x and
the various stages may therefore be shown thus:
(a) 2x x 3 =6x
(b) 6x+2 =3%
() 3xx3 =9x
(d) 9x+9 =X
(i1) If the original number is odd, it is of the form 2x 1.
(a) (2x+1) X3 =6x+3
(6) (6x+43)+2 =gx+1}
add % =3x+2
(c) (3x+2) x3 =9gx+6
(d) (9x+6)+9 =x+3
EXAMPLE 2
(a) Select a number and multiply it by 5.
(b) Add 6.
(¢) Multiply by 4.
(d) Add 9.
(e) Multiply by 5.

The result of these operations is made known; the original
number is easily found by deducting 165 from the result,
giving a number ending in two noughts. The digits to the left
of these noughts form the original number. In other words,
the result minus 165 is equivalent to one hundred times the
original number.

Thus:

Selected number:
Multiply by 5:
Add 6:

Multiply by 4:
Add o: