#### Is it worth mentioning this aspect of some recreational math problems?

There are several river crossing recreational math problems that have an interesting common feature that I have never seen anybody talk about. Here is a sampling of such problems up to the frog jumping puzzle.

All these problems involve using a boat to move people and things from one side of a river to the other. Suppose we have a solution to one of these problems and imagine filming the execution of the winning moves. Now imagine playing the film in reverse. We see the moves in reverse order and moving in the opposite direction to take everyone from the end position to the start position.

If you think about it for a moment, you will see that we can apply this to the original problem. If we put everyone back in the starting position and do the original moves in reverse order, we end up with everyone in the end position, which is another solution.

If I still have your attention, for most of these problems it gets even more interesting. If you start to solve the problem, you notice that there is one move that creates the mirror image of the previous position. That is, if the boat and group A are on one shore and group B on the other, after crossing you have group A and the boat on the opposite shore and group B on the original shore. Reasoning as before, what this means is that if you do all the moves before the last in reverse order, you end up with a solution to the problem.

To apply this idea to some of the problems, you have to see certain equivalences. In the first problem with the goat, cabbage and fox, you have to realize that the fox and cabbage are equivalent. Neither one can be left alone with the goat.

For the Japanese problem before the frogs, you have to realize that the father and two sons are jointly equivalent to the mother and two daughters. For any position, if you swap mother/father and son/daughter you have an equivalent setup.

Does what I said make sense? I had to get that out of my system. It just irks me that nobody ever mentions this. It turns out these same ideas can be applied to the frog jumping puzzle, but that will have to be for another day.

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