Maybe the real treasure was the friends we made along the way

"Here's a  puzzle for you," says Bob. "An owner of a dog, a cat, and a hamster needs to cross a river, and there is one rowboat on their side of the river.
"The owner must be onboard to row the boat, but must be careful who is left alone (on either shore).
"If the dog is left alone with the cat, they will attack each other; if the cat is left alone with the hamster, the cat will eat the hamster.
"Given the size of the rowboat, the owner is only able to bring one pet on the boat at any time.
"Can the pet owner get everyone across the river safely?"

Alice wrestles with the puzzle for a while. "I'm stumped!" she professes. "What's the solution?"

Bob grins. "The answer to the puzzle is 'yes.'"

Mathematics is a rather unpopular subject in school, and likely for many reasons. It starts out simple, but very monotonous (maybe addition was a fun little exercise… until you had to do hundreds of examples of long addition; maybe multiplication was neat… until you needed to memorise the multiplication table for the first ten numbers or so). Then, things get more abstract and complicated. You learn about a bunch of “rules” of algebra, and strange notation (and letters!), and it can be tough to wrap your head around everything (especially when it doesn’t seem useful anymore).

It only gets worse when your teacher starts to implement artificial handicaps. You learn how to use a calculator, and then suddenly it’s not allowed for exams? Maybe you’re in calculus and learn l’Hôpital’s rule—and then the exam says outright that “you may NOT use l’Hôpital’s rule,” solely because it makes the problem too easy otherwise! Worst of all, maybe you’ve spent hours upon hours trying to wrestling with homework, and when you finally figure it out and submit it, your teacher starts deducting points for “unclear work” even though your final answer is correct!

Even from the other side, perhaps you’ve really mastered the material and don’t need to write very much to solve the homework problems (for example, maybe you can multiply those large numbers in your head, or maybe you can calculate the derivative “in one go”). This should be met with praise, but instead your teacher gives you a big X and writes “Show your work!”

In grade school, I wasn’t one of those kids who struggled with arithmetic, the rules of algebra, or anything like that. I was even supportive of the absence of calculators, perhaps because it gave me an edge over my classmates. (I never said I was an angel growing up.) Instead, I was one of those kids who tried to solve problems in as few steps as possible, minimising the amount of work needed to get to the answer. I thought the need to show my work when solving an algebra equation was an insult to my intelligence, because I didn’t need all that hand holding to figure the problem out! I learned the ropes for exactly how much work was necessary to get full marks, and stuck to that rope like glue.

When I enrolled in university—as the final jewel to my crown of mathematical superiority—I enrolled in the honours mathematics courses, despite being a computer science student and having no need to go the extra distance mathematically. I had no idea what the difference between honours and ordinary maths courses was, but surely this was going to be significantly harder, and that was the whole point.

Okay, maybe I’m being a little hyperbolic. Sure, I was an arrogant kid, but I also just really enjoyed doing mathematics, and spent a lot of time messing around with it for fun. Math had so many cool symbols (like , or ) and cool concepts (like different kinds of infinities, like or ), and I just wanted to be able to understand them. Perhaps ironically, the Wikipedia articles on these things were way too complicated for me to understand, though, so it would be a while before I actually would come to understand these things. Unlike young mathematical talents who are motivated and intelligent enough to start actually getting into real mathematics while still in high school, I would remain wet behind the ears until well into my university career.

When I entered the honours maths courses, my expectations were shattered. I had done calculus in high school, and my dad had given me a linear algebra textbook to work through back then as well, so I thought I was more than prepared for what was to come. To my surprise, it turned out that mathematics was nothing like what I had learned it to be up until this point. We wouldn’t see derivatives in our calculus class (which is the main object of study for calculus!) until halfway through the term, and we wouldn’t see matrices in our linear algebra class (the main object of study for linear algebra!) for a good few weeks or so either.

I had expected the honours courses to pick up the pace significantly from an ordinary maths course so that we could get straight to the cool, really complicated mathematics, but I was greatly mistaken. In fact, we were going to move slower. If I wanted to see a broader range of advanced mathematics topics as quickly as possible, I should have taken the engineering maths courses instead.

What was slowing the honours maths courses down so much? Surely, the only students enrolling in honours maths were those who already had an inclination for mathematics, so why were we losing to the engineers? The answer is that the professors had to make us unlearn a lot of what we already knew, and build us back up properly. The first few weeks of both classes was dedicated to mathematical formalism and rigour. Even though we may have all been familiar with calculating limits from high school, we didn’t get to doing that in university until the limit was defined carefully and its properties were proven rigorously. The homework seldom asked us to calculate answers anymore; instead, usually the answer was already given in the question!

To illustrate, instead of being asked to calculate the derivative of , we would instead be asked to prove that the derivative of was equal to . The answer is provided in the question, so clearly the answer wasn’t important to begin with.

You may have been wondering where my personal anecdote was going, but now we’ve made it. In mathematics, it’s not about the answer! Sure, the answer is important, but no mathematician would ever be content with just the answer. For instance, many of the Millenium Prize Problems (quite literally the “million dollar questions” in mathematics) have “known” (that is, widely accepted) answers, but the challenge is to actually prove that the answers are correct.

Since you have patiently sat through my meandering anecdote for such a meagre punchline, let me offer an olive branch in the form of a joke:

An engineer, a physicist, and a mathematician are each locked in individual prison cells.

One night, the engineer is awoken by a sudden fire in his cell. He looks around and finds a pale of water nearby. He dumps the entire pale of water on the fire, and the fire goes out. He goes back to sleep peacefully, knowing the fire is out.

One night, the physicist is awoken by a sudden fire in his cell as well. He notices the pale of water nearby, and calculates that he only needs two-thirds of the water to put out the fire. He puts out the fire, and goes back to sleep peacefully, knowing that the fire was put out efficiently.

One night, the mathematician is awoken by a sudden fire in his cell. He looks around frantically for a solution, and notices the pale of water nearby. He does nothing but smiles and goes back to sleep peacefully, knowing that the pale of water would be enough to douse the fire.

I don’t remember who told me this joke

When learning mathematics in school, you definitely get inundated with notations and formulae and particular approaches to solving problems, and this can make it easy to lose sight of what mathematics is about: it’s a form of logical and creative thinking geared towards solving precise problems. The formulae and rules you learn in school are there to showcase what creative thinking has already been done in the past, and help you use them to build your own problem-solving toolkit. People tend to think that mathematics is about multiplying huge numbers, or calculating really complicated integrals, but the real mathematics was more the development of a method for long multiplication, or the formalisation of integrals in the first place! However, you need to learn how to walk before you can run, and likewise you need to learn how to perform calculations before you can start deriving new mathematical theories.

Let’s circle back to why your maths teachers (should!) care that you show enough work. Sure, from an educational point of view, you showing your work would demonstrate to your teacher that you understood the techniques taught in-class (and didn’t just copy the answer from a friend), but there’s a bigger vision here too. The most important reason (in my opinion) ties back to mathematics being the development of problem-solving skills. Ultimately, the mathematical community is collaborative. Showing work is an opportunity to practise your mathematical communication skills: your work is not only for you, but it’s so you can help others understand how to solve the problem as well. Yes, perhaps you are currently explaining how to solve the problem to your teacher, who likely already knows how to solve the problem as well, but down the road you might come up with something ingenious, and at that point we can only hope you have the necessary skills to share your brilliance with the rest of the mathematical world.

P.S. If you’ve made it this far, and are wondering what the actual solution to Bob’s puzzle is, I will describe how the pet owner brings all of the pets across the river safely below. Let’s say that, at the beginning of the puzzle, the pet owner and the pets are all on side A of the river, and they want to finish on side B. This isn’t the only solution.

• First, the pet owner brings the cat to side B. This leaves the dog and the hamster on side A, which is not a problem.
• The pet owner then rows back to side A alone. Then, the pet owner brings the hamster to side B. Now, the dog is left alone on side A, and side B has the cat and the hamster. Note that the cat doesn’t eat the hamster, though, because the pet owner is also on side B.
• The pet owner brings the cat back while rowing to side A. This leaves the hamster alone on side B, and the dog and cat are together with the owner on side A.
• The pet owner now brings the dog to side B. There are no conflicts: the cat is alone on side A, and the other pets have no qualms.
• Now, the pet owner rows back to side A alone. The cat is the only pet left on side A, so the pet owner brings the cat to side B. Finally, all of the pets (and the pet owner) are on side B!

As a table, here is the solution. The characters are: the pet (O)wner, the (D)og, the (C)at, and the (H)amster:

Move	Side A	Side B
	ODCH
owner brings cat to B	DH	OC
owner returns to A	ODH	C
owner brings hamster to B	D	OCH
owner brings cat to A	ODC	H
owner brings dog to B	C	ODH
owner returns to A	OC	DH
owner brings cat to B		ODCH