Mathematical_Recreations-Kraitchik-2e

Numerical Pastimes 87
if a player A occupies a strategic position, his opponent B
cannot capture a strategic position on his next play, but A
can obtain a new strategic position on his next play, regardless
of what B does.
The winning principle in nim rests on the following theorem:
Let the numbers of a position (a, b, c) be written in the
binary scale of notation. Then the position (a, b, c) is strategic
if and only if the sum of the digits in each place is even. (For
example, (5, 43, 46) is strategic since
decimal 5 = binary 101
decimal 43 = binary 101,011
decimal 46 = binary 101,110
202,222
It is to be noticed that we do not "carryover" from one
column to the next; only the sums of the individual columns
are considered.)
First we shall show that every position which can be
reached in one play from a position of this type is not of this
type. For by the rules just one of the piles or numbers a, b, c
is changed. Such a change produces in the binary notation
of the number changed a redistribution of the digits ° and 1;
that is, at least one 1 must become a ° or one ° must become
a 1. Hence in at least one column the sum of the digits is
changed by 1, that is from an even number to an odd.
Next we show that in one playa position which is not of
this special type can be made so. Find the first column from
the left for which the sum of the digits is odd, either 1 or 3.
If this sum is 1, then just one of the three piles or numbers
has a 1 in this place, and that is the number which must be
changed; if the sum is 3, each of the three piles has a 1 in this
place, and anyone of the piles may be changed. Having decided
which number is to be changed, replace the 1 in this
position by a ° and alter the digits following it by interchang-