Does this explanation related to recreational mathematics river crossing problems make sense?

I don’t know if it makes sense. I have a hard time caring about an arbitrary definition of complementary positions, and you have skipped what to me are some necessary definitions about what you consider a state, and what solution you are thinking of.

As with many story problems, I take them as actual problems in reality, not as math, and have a hard time dealing with the math-i-fication. I always get excited hearing about an actual situation with people in a dangerous situation, and then get extremely disappointed by the assumptions of the problem, and the lack of explanations, and then the lack of the credibility of the situation and the explanations.

For example, I wonder who is in charge, and what the behavior of each person is. It’s a little hard for me to accept the situation as stated, because who is going to coordinate these movements?

Also it seems like you and the problem are skipping explaining how the actual loading and unloading works, and what moves actually work or not.

For example, it seems to me there’s a state where you’ve got someone taking the boat back, and I’m not clear whether it works to send a cannibal back with the boat, and if so, whether he can be trusted not to disembark on a side with equal cannibals and missionaries, or whether he’ll politely obey the problem-solver’s orders to stay in the boat.

I am guessing that your notion of complementary positions is based on not counting the intermediary states where there are people in the boat on neither side?

Since you seem to think there’s one solution, I’m guessing it’s probably something like this:

CCCMMM~~~~~______
__CCMM~CM>~______
__CCMM~~<C~M_____
___CMM~CC>~M_____
___CMM~<C~~CM____
____CM~CM>~CM____
____CM~<C~~CMM___
_____M~CC>~CMM___
_____M~<C~~CCMM__
______~CM>~CCMM__
______~~~~~CCCMMM

And, ok, there you can see the visual symmetry. It could be even clearer by arranging the C’s and M’s in opposite order.

On the other hand, I think there’s at least one variation, where the first return trip is done by a missionary:

CCCMMM~~~~~______
__CCMM~CM>~______
__CCMM~~<M~C_____
___CMM~CM>~C_____
___CMM~<C~~CM____
____CM~CM>~CM____
____CM~<C~~CMM___
_____M~CC>~CMM___
_____M~<C~~CCMM__
______~CM>~CCMM__
______~~~~~CCCMMM

Which is almost the same thing, but isn’t symmetrical.

You are correct about having other solutions. There are two possibilities for the first two moves, which means there are two possibilities for the last two moves, giving two times two equals four possible solutions.

Would it be clearer if we speak of mirror image instead of complements?
One thing that I left out of the definition that needs to be included is the position of the boat. In doing a crossing to get a mirror image of the position, the boat will also be on the opposite side.

Mirror image relates to the symmetry that you mentioned.

@LostInParadise Would it be clearer if we speak of mirror image instead of complements? Yes, I’d talk of mirror images and symmetry, and draw a diagonal line through the diagram, or even place an actual mirror across a page with such a diagram, and invite each student to see.

“Complementary” is a mathematical term, and good to know, but unless the students are into Latin word roots, doesn’t communicate – also because it could apply to many perspectives – complementary in what sense? Well, only really if you visualize the positions over time as I did above, or if you can follow a series description like you initially gave (and I bet that’s much less excessive except to really math/function/concept-oriented students, who I’d expect to be either the minority, or to be able to get the idea very easily). If thing being complementary this way is a teaching goal of the lesson, then I’d be interested to see more examples, and why the observation is useful.

The thing that has always thrown me about this problem, since I first encountered it about 1980, is that how the boat works isn’t really spelled out well enough, nor is the behavior of the two types of people.

In particular, since I think it’s pretty much necessary to send a cannibal as the only type in the boat at multiple points in time, I tend to think of the cannibal choosing what to do (being human), and I see a couple of opportunities where a cannibal with free will and a boat, could choose to disembark rather than just pick up a passenger, or turn around and disembark, so that they could outnumber and eat a missionary or even two (at which point, then could also then sail to the other side and eat the missionaries there. And if they’re people who would eat a missionary given the chance, it seems putting them in a boat with no missionaries, gives them the opportunity, so why don’t they do that?

And that’s one of the main moral violations of story problems, in that they claim to be showing how a situation in real life relates to math or logic and homework/test exercises which supposedly only have one valid interpretation and one correct answer, but they really don’t, and the official answer really isn’t the most correct one. As such, it’s kind of a major violation of trust between teacher and student, since the teacher is supposed to be helping the students to understand things better, but in these cases, they’re demonstrating that teachers have the power in our system to violate the truth and insist students surrender their own right understandings, in order to not be penalized and told they are wrong.

What I’d really want, would be for the teacher to be listening for such correct perspectives by students, and readily admit that the student is very correct (understanding a problem outside its box would earn my highest grade), and then explaining what the problem’s author was thinking, and what their unwritten rules are.

I don’t see where the cannibals would have a chance to eat the missionaries. The only difficulty I see is that, at the beginning of the problem, if there are two cannibals that row across the river, they might run away. In the case where you start with a missionary and a cannibal, the cannibal might run away while the missionary is returning the boat. To prevent these situations, it can be said that if they occur, the missionaries, outnumbering the remaining cannibals, would threaten to take revenge.