Foxes And Geese
(
MISCELLANEOUS PUZZLES)
Here is a little puzzle of the moving counters class that my readers will probably find entertaining. Make a diagram of any convenient size similar to that shown in our illustration, and provide six counters—three marked to represent foxes and three to represent geese. Place the geese on the discs 1, 2, and 3, and the foxes on the discs numbered 10, 11, and 12
Now the puzzle is this. By moving one at a time, fox and goose alternately, along a straight line from one disc to the next one, try to get the foxes on 1, 2, and 3, and the geese on 10, 11, and 12—that is, make them exchange places—in the fewest possible moves.
But you must be careful never to let a fox and goose get within reach of each other, or there will be trouble. This rule, you will find, prevents you moving the fox from 11 on the first move, as on either 4 or 6 he would be within reach of a goose. It also prevents your moving a fox from 10 to 9, or from 12 to 7. If you play 10 to 5, then your next move may be 2 to 9 with a goose, which you could not have played if the fox had not previously gone from 10. It is perhaps unnecessary to say that only one fox or one goose can be on a disc at the same time. Now, what is the smallest number of moves necessary to make the foxes and geese change places?
Answer:
The smallest possible number of moves is twenty-two—that is, eleven for the foxes and eleven for the geese. Here is one way of solving the puzzle:
10—5 |
11—6 |
12—7 |
5—12 |
6—1 |
7—6 |
—— |
—— |
—— |
—— |
—— |
—— |
1—8 |
2—9 |
3—4 |
8—3 |
9—10 |
4—9 |
12—7 |
1—8 |
6—1 |
7—2 |
8—3 |
—— |
—— |
—— |
—— |
—— |
3—4 |
10—5 |
9—10 |
4—11 |
5—12 |
Of course, the reader will play the first move in the top line, then the first move in the second line, then the second move in the top line, and so on alternately.
In A. in M., p. 230, I have explained fully my "buttons and string" method of solving puzzles on chequered boards. In Diagram A is shown the puzzle in the form in which it may be presented on a portion of the chessboard with six knights. A comparison with the illustration on page 141 will show that I have there dispensed with the necessity of explaining the knight's move to the uninstructed reader by lines that indicate those moves. The two puzzles are the same thing in different dress. Now compare page 141 with Diagram B, and it will be seen that by disentangling the strings I have obtained a simplified diagram without altering the essential relations between the buttons or discs. The reader will now satisfy himself without any difficulty that the puzzle requires eleven moves for the foxes and eleven for the geese. He will see that a goose on 1 or 3 must go to 8, to avoid being one move from a fox and to enable the fox on 11 to come on to the ring. If we play 1—8, then it is clearly best to play 10—5 and not 12—5 for the foxes. When they are all on the circle, then they simply promenade round it in a clockwise direction, taking care to reserve 8—3 and 5—12 for the final moves. It is thus rendered ridiculously easy by this method. See also notes on solutions to Nos. and .
Informational
