Mathematical_Recreations-Kraitchik-2e

90 Mathematical Recreations
but not from (2) to (3). Position (4) can be obtained in two
ways: (a) the fast way, taking the last two rings from (1) at
the same time; (b) the slow way, from (1) to (3), and then
taking off the last ring. Position (5) can be obtained from
(4) in one step.
The following tables give the successive positions assumed
in taking all the rings of a 7-ring toy off the bar by the
two methods:
First: The Slow Way
••••••• 000000 0 0 00 o 0 • ••••••
Second: The Fast Way
0 00000 0 00 00 0 0 000000 0 00 0
00 0000 0 000 00 00 00 0000 0 00 00
0 00000 0 00 00 000 0 00000 0 0&8 00 0
00 000 0 0 0 000 000 000 000 0 0 0 00 000
000 000 0 0 0 000 00 0000 000 0 000 000
80g
0000 0000 0 000 000 0 00 00000 0 000 0
000 00000 0 00 000 00 000 00 00 0 00 000 0 0
00 00 00 0 000 000 0 0 0 00 0 0000 000 0
000 00 0 0 0000 000 0 00 0 00 0000 000
0 0 00 00 0000 000 0 0 00 0000 00 00 000 00
0 00 0 00 000 000 00 00 00 000 00 00 I) 0000 00
0 0 0000 00 00 0000 00 00 00000 00 0 0 0000
0 00 000 00 00 0 0000 0 00 0 0 000 000 0000 0
00 00 0000 000 0 0000 000 0 0 0 00000 000000
000 00000 000 0000 0 000 0 o 0 00 00 00000
00 0 000 00 0 0 00000 0 00000 0 0 00 0000000
00 0 0 00 00000 00000
000 0 0 0 00 00 00000o
000 0 00 00 0 00000oo
0000 0 0 00
00000 0 00 0
M. Gros gave a very simple rule for counting the number
of steps needed to pass from one position to another. In
order to derive this rule we shall use his notation for the positions.
For a toy with n rings each position is denoted by an
n-digit number in the binary scale (including numbers which
begin with one or more O's), each place in the number being
reserved for a symbol, 0 or 1, to denote the position of a particular
ring. To those places in the binary number which
correspond to rings on the bar are assigned alternate l's and
O's, starting with a 1 at the left and disregarding the other