Reversing a Puzzle

It frequently happens that a puzzle consists of obtaining a given final position from a given initial position by making the smallest possible number of moves. One can then starts from the final position instead, and try to obtain the initial position by backward moves. If this has succeeded, all one has to do to get the solution of the original puzzle is to perform all moves in the reverse order and in the reverse direction. It is virtually certain that the new puzzle (the reversed one) will be simpler; otherwise the puzzle would surely have been presented in the reversed form. Hence, in most cases the reversal of the puzzle will produce a simplification.

Example:  Golf and Tennis

Three golf balls (the small circles) and two tennis balls (the larger circles) are placed in a row, with golf balls and tennis balls alternating (Fig-initial). Now with every move a golf ball and an adjacent tennis ball are moved as one whole, so that they remain adjacent. You are not allowed to reverse the balls; that is, you should not interchange the golf ball and the tennis ball. Moving the golf ball and the tennis ball is considered as one move. The problem is to get the golf balls and the tennis balls in a row in such a way that the golf balls are adjacent, and the tennis balls, too (Fig-final), in as few moves as possible.

 As rotation is not allowed, the new row is parallel to the original row. However, it is not necessary that the golf balls get to the left of the tennis balls, as in the Fig-final; as a matter of fact, if you have managed to get the golf balls on the left-hand side, all you have to do to get the golf balls to the right of the tennis balls is to perform the mirror image of each move. If it were allowable to interchange a golf ball and an adjacent tennis ball, the final position could be reached without trouble in three moves; the relative order of golf balls would not have changed, any more than that of the tennis balls. Since interchanging a golf ball and a tennis ball is forbidden, it is slightly more complicated to reach the final position.

Solution :

In the initial position, in which these are four cases where a golf ball and a tennis ball are adjacent, a large number of moves can be made. However, from the final position only two moves are possible. Therefore we reverse the puzzle and consider the final position as the initial one; in the new initial position we designate the golf balls and tennis balls as in Fig-num

In determining the moves, one should observe that there is no point in performing two consecutive moves with the same pair (because then one can combine both moves into one). Furthermore, one should take care that after every move there will be another pair (consisting of a golf ball and an adjacent tennis ball) with which a next move can be made. This leads to the following puzzle tree

­The dots indicate free positions. From the position 3. . 2A 1 B infinitely many moves are possible. You can place the pair 2 A (or I B) wherever you want, because in each case you can make another move with the pair IB (or 2A). However, we have only included those third moves (following 3 . . 2A 1 B) after which it is possible to make a fourth move which yields the desired final position (the original initial position). Hence, this can be done in four ways; in the first two moves these agree completely and for the rest they come to practically the same thing: the last two moves take the pairs 2 A and 1 B to the left of3, the order being unimportant, while it also makes no difference which of these pairs gets next to 3. The four ways are:

then the solution in question is