Mathematical_Recreations-Kraitchik-2e

Numerical Pastimes 91
rings. If the first ring from the left is off, the digit 0 is assigned
to this ring's place in the number. If any other ring
is off, we assign to its place the same digit which the ring to
its left has; that is to say, the digit to the left of its place is
repeated. In this way the set of all positions corresponds
uniquely to the set of all numbers from 0 to 2" - 1, inclusive.
The table compares the notation for four arrangements.
No.
1
2
3
4
Position of rings (* = on, 0 = off)
*0**000
*00*000
*0**00*
*0**0 **
Cros notation
1101111
1110000
1 101 1 1 0
1 101 101
It can easily be shown that, starting from any position, the
operation of taking off a ring which is in position to be taken
off either increases or decreases the corresponding number by
1, and similarly for putting on a ring. Conversely, if the
number corresponding to a given position is greater than 0,
then the position corresponding to the next smaller number
can be obtained by putting on or taking off one ring; and in
like manner a number less than the maximum can be increased
by 1. Since the object of the game is to reduce the
number 10101· .. to 00000· .. in the fewest possible number
of steps, this may be accomplished in just 10101· .. operations.
According to whether n is even or odd this number is
t(2,.+l - 2) or t(2 n + 1 - 1). These results are for the slow way.
To find the number of steps by the fast way note that we
save 2 steps out of every 8; except at the end, where we save
1 step out of 2 if n is even, 1 step out of 5 if n is odd. From
this we find that the number of steps is 2,.-1 - 1 if n is even,
2 ..- 1 if n is odd.
4. THE TOWER OF HANOI. The theory of the Chinese