THE EXCHANGE PUZZLE.
(
Moving Counter Problem)
Here is a rather entertaining little puzzle with moving counters. You
only need twelve counters--six of one colour, marked A, C, E, G, I, and
K, and the other six marked B, D, F, H, J, and L. You first place them
on the diagram, as shown in the illustration, and the puzzle is to get
them into regular alphabetical order, as follows:--
A B C D
E F G H
I J K L
The moves are made by exchanges of opposite colours standing on the same
line. Thus, G and J may exchange places, or F and A, but you cannot
exchange G and C, or F and D, because in one case they are both white
and in the other case both black. Can you bring about the required
arrangement in seventeen exchanges?
It cannot be done in fewer moves. The puzzle is really much easier than
it looks, if properly attacked.
Answer:
Make the following exchanges of pairs: H-K, H-E, H-C, H-A, I-L, I-F,
I-D, K-L, G-J, J-A, F-K, L-E, D-K, E-F, E-D, E-B, B-K. It will be found
that, although the white counters can be moved to their proper places in
11 moves, if we omit all consideration of exchanges, yet the black
cannot be so moved in fewer than 17 moves. So we have to introduce waste
moves with the white counters to equal the minimum required by the
black. Thus fewer than 17 moves must be impossible. Some of the moves
are, of course, interchangeable.
