River Crossing

Math at my age is a rather complicated thing. It isn’t particularly challenging in the grand scheme of mathematics but many of my peers often hear the word “math” and their immediate reaction is to grunt or sigh. I, however, have a slightly different reaction to that. I find myself looking at math outside of class and learning more about it outside of the school setting one such piece of mathematics that I will look into in my free time is math puzzles. One such puzzle is a well know one e that goes by many names, including the river crossing puzzle, wolf, goat, cabbage problem, the boatmen problem, and many other variations of the puzzle. As I am interested in these things I have talked to many math teachers about this puzzle and a few others. The thing that I find fascinating is many of the teachers I will strike up a conversation with about this topic will tell me about a puzzle they learned when they were my age or a little younger that got them interested in math and how the answer to the puzzle has stuck with them since they learned it. I would like to discuss a few of my ideas on the puzzle and what I find interesting about it specifically and how it has become a puzzle I will never forget.

Now to the actual puzzle, the premise of which is relatively simple. The story goes that you are a farmer who just came back from the market having bought three things(this is where the puzzle deviates and has many substitutes but I will give just one example) you have a wolf, a goat, and some cabbage. The issue is on your way you realize that you have to cross a river to actually make it home. You want to make sure you keep your wolf, goat, and cabbage. There is a boat that sits at the banks of the river and can fit you and one of your other items and only one. The other issue you face as you begin untieing the boat is that in taking one item across you naturally will have to leave two behind and you aren’t explicitly worried about your animals wandering off but you do know that if you leave the wolf and the goat on the same side of the river unattended then you will no longer have a goat. If you leave the goat and the cabbage unattended, the goat will eat the cabbage. Now the question is, how do you get all three of your items across the river safely?

(image found at ResearchGate.net)

This puzzle is easy to get stumped on because it feels like no matter how hard you try you always need help to solve it. I won’t quite reveal the answer but I will discuss what I find interesting about it. Firstly, I find it very interesting how versatile it is as a whole. It doesn’t have to be a goat, cabbage, and wolf, it can be a goose instead of a goat and a bag of grain instead of the cabbage, and a fox instead of a wolf. It can be just about anything you want and it’s easy to find all sorts of situations where the same rules apply just by switching around a few names. Secondly, I enjoy how the idea of the puzzle doesn’t feel like it’s a very distant idea. It can be explained with items that everyone knows what they are and doesn’t require explaining how some super high-tech device is needed. It is a relatively easy concept to explain and uses only things that everyone knows about. Thirdly, the puzzle is best described and most easily attempted with an illustration. It is easiest to see all the parts of the puzzle if you can draw a farmer who has a cabbage, goat, and fox in tow and approaches a river with a boat where you can then draw lines and find any way you want to illustrate the movement of your items. Fourth, is the story that ties the puzzle together. If there are all sorts of ways to tell the story you get to choose the version that interests you the most despite them all being practically identical. We are best able to understand the puzzle with the simplification of using terms that we all know. The puzzle has been studied quite a bit in all sorts of fields of math and could be described using terms that you would only know if you are interested enough to learn the terms and concepts associated with it.

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Now for the solution and the last thing that I enjoy about the puzzle. If you haven’t already taken a few shots at the puzzle I would recommend it because the trick to the puzzle will make you wonder why you never thought of it. First, your only option to not lose immediately is to take the goose across leaving the wolf and cabbage alone. Then, you will go back to another side of the river empty-handed. Next, you will take the wolf over. Here is where it gets tricky. You will take the goose from the opposite side of the river and bring it back with you leaving the wolf on one side, the cabbage on the other, and the goose in the boat with you. This is where my last big note of what is neat about the puzzle in my opinion. Nothing about the puzzle said you couldn’t take anything back with you after having brought it to one side. To finish the puzzle you will drop off the goose and take the cabbage across once again leaving the cabbage and the wolf alone. Now go back empty-handed once again and take the goose over to finish bringing all of your items across the river. It is also a fun part of these puzzles in my opinion to have that instance where you realize the trick to it. In this case, it is that you are allowed to traverse with your items going both ways on the river. I remember when that clicked after being told the answer and just being stunned at how I didn’t think of it earlier.

(Obviously, this is a slightly different variation but you can get the idea. in these diagrams, the cabbage is the carrot but all other rules still apply. Images found at transum.org where you can play various variations at the game and will be linked at the end so you can try it yourself)

First step. Bring the goat over to the other side.

Next, bring the wolf over

Bring the goat back over

Bring the carrot(cabbage) over

lastly, bring the goat back over and you have done it.

this is one of many puzzles I find fascinating and intriguing, but it displays many of the pieces that I think are important to what I wanted to discuss here. This is a unique take on storytelling; it isn’t a novel, so you don’t get to be introduced to characters and feel them on a personal level. It isn’t a poem where the symbolism might drive the story and the message of the story home. It is just a puzzle at its core but where I think it gets really interesting is that you can always describe these things using complicated math jargon or in really abstract terms. We can make sense of it though as an average person because it explains it in terms that generally speaking we all know and understand. We get to imagine how the farmer got to the situation he is in and why maybe he is only getting the three items he did get. It even has a plothole. If you stare at the piece of paper you drew the puzzle down on or read it enough times you might catch that there is an implication that no matter what he would have to cross the river to get back to his home so maybe we can begin to imagine that this farmer weirdly enough doesn’t remember where his house is or is just an awful planner and didn’t for some reason foresee any problems in only having a small boat to row back and forth to get back to his home. We can begin to formulate the pieces of the story that are “missing” and might be included in other styles of stories just by indulging ourselves and just enjoying the ride of the story. I know I didn’t illustrate the story in the best way and you can get plenty of other sources that do it better but I would like to make you think more so about how interesting it is in the first place that we get to do things like imagine what it is that got the farmer to this situation and how maybe we can learn from this that there might be a point where you have to reverse a step you already took to get to your ultimate goal.

Other places to learn more about this specific puzzle:

River Crossing Challenges