Mathematical_Recreations-Kraitchik-2e

58 Mathematical Recreations
columns those numbers whose ternary notation has a 1 or a
2 in the last place; in the third and fourth columns, those
having a 1 or a 2 in the next-to-the-Iast place, and so on. Or
we may use negative digits, and list in a single column those
numbers -having a 1 or a -1 in a given position, distinguishing
those which have a -1 by an asterisk. In the table at the
right, page 57, a number such as 18 or 24 that is to be formed
by subtraction from 27, is characterized by the fact that it
appears with an asterisk in the first column from the left in
which it is listed.
The binary system has many other applications. In §12
of this chapter we shall show its application to the game of
Nim and to the theory of the Chinese rings.
3. TO GUESS A SELECTED NUMBER
There are innumerable ways of finding a number chosen
by someone, provided the result of certain operations on it
is known. We shall confine ourselves to certain typical
methods in ordinary use.
1. Ask the person who has chosen the number to triple
it and tell you whether the resulting product is even or odd.
If it is even let him divide it by two; if odd, let him add one
and divide by two. Now request him to triple the result of
the second step and then tell you how many times nine goes
into the result, disregarding any remainder. If x is his final
answer then 2x or 2x + 1 is the number he selected, according
as the result of the first operation was even or odd.
The demonstration of the method is easy. If the number
selected was even, say 2x, then we have successively: 2x·3
= 6x; 6x + 2 = 3x; 3x·3 = 9x; 9x -;- 9 = x. If the number
selected was odd, say' 2x + 1, then (2x + 1)3 = 6x + 3;
(6x+3+1)+2=3x+2; (3x+2)3=9x+6; (9x+6) +9
= x with a remainder 6 to be discarded.