What Is Mathematics? An Example Using Dominoes

You don’t need any specialist maths knowledge to read this post. A secondary-school understanding of basic arithmetic is more than enough.

Lots of people, including my own family, ask me what it’s like to study maths at university. They want to know which modules I took, whether I spent all day doing calculations, or what a mathematician actually does.

In this post, I won’t go through university-level maths topics one by one. Instead, I’ll focus on a more fundamental question behind all of these: how closely does the maths we’re taught at school reflect the true nature of mathematics?

At school, maths generally revolves around learning specific methods and applying them to the right kinds of questions: solving equations, taking derivatives, calculating integrals, or using the formulae we learn in class.

Being able to do these things is, of course, important. But maths isn’t just about choosing the right method and getting the right answer. Making definitions, looking at examples, forming conjectures, searching for counterexamples, and proving why a result is true are also fundamental parts of maths.

Paul Lockhart’s A Mathematician’s Lament explores the same issue much more fully—and more sharply—arguing that school often reduces maths from a creative activity to a list of rules and procedures.

Why do we study maths?

When people ask why maths is taught at school, the usual answer is that it’s needed for later stages of education and certain jobs, rather than something we use directly in everyday life. In many countries, maths is presented as preparation for later subjects, exams, and particular career paths.

Maths does have important applications in engineering, technology, finance, economics, architecture, transport, science, and many other fields. But if we see maths education only as preparation for later subjects, exams, or professional use, the fundamental parts of mathematical thinking—asking questions, spotting patterns, making conjectures, and proving things—can fade into the background.

After years of maths education, what often remains is a handful of half-remembered formulae and, all too often, a negative impression of maths itself.

So, what is doing maths like?

Mathematicians do calculations and use different methods, of course. But the real point often isn’t to apply a ready-made method. It’s to work out which questions are worth asking and why a claim is true or false.

Now let’s try out this way of thinking with a little game involving dominoes.

Definitions for the domino problem

Let an n × m board be a rectangular grid made up of n rows and m columns.

A domino is a 1 × 2 rectangular piece. When placed on the board, it covers two squares that share an edge. It can be placed horizontally or vertically, but not diagonally.

In the diagrams below, each pair of adjacent squares of the same colour represents a single domino. The colours are used only to make the different dominoes easier to distinguish.

Tiling a board with dominoes means covering every square on the board exactly once. No squares are left uncovered, no dominoes overlap, and no domino extends beyond the board.

Example

Can we cover a 4 × 4 board with dominoes?

Yes. The sixteen squares below are grouped into eight same-coloured pairs. Each pair is one domino, so the board is completely tiled.

Now let’s move on to our first question.

Question 1

Imagine the square in the top-left corner of a 4 × 4 board has been removed. It is no longer part of the board and cannot be covered. Can we cover the remaining area with dominoes?

Question 2

This time, suppose the squares in the top-left and bottom-left corners have been removed. Can we cover the remaining area with dominoes?

Question 3

Now suppose the squares in the top-left and bottom-right corners have been removed. Can we cover the remaining area with dominoes?

What makes a solution elegant?

What’s really surprising here isn’t that the board can’t be covered. It’s that we can understand why without trying every possible arrangement one by one.

At first, colouring the board black and white might seem unrelated to the problem. But this creative point of view brings the key property into focus: every domino has to cover exactly one square of each colour. One short observation rules out every possible arrangement at once and shows us why the result is true.

The same result can be proved in different ways. Even if all the proofs are correct, one might be shorter, more general, more surprising, or make the central idea clearer. That’s why mathematicians might call a solution “elegant” or “beautiful”. Perhaps this is exactly what makes maths a little like art: coming up with unexpected, illuminating ideas within a set of rules.

The fact that maths is used in engineering, technology, science, and many other fields doesn’t make it any less beautiful. Seeing it only as a tool other subjects need means missing a big part of it. Sometimes we ask a question simply because it’s interesting. Sometimes a proof matters simply because the idea it reveals is beautiful.

That’s why a good maths education shouldn’t stop at teaching methods. It should also show what it feels like to make a conjecture, doubt it, look for a new point of view, and be surprised by a short proof. Even if we forget most of the formulae years later, we can still remember that maths is a way of thinking.

A few questions for discussion

Learning maths doesn’t have to be a solo activity. Discussing ideas, coming up with examples together, and questioning each other’s explanations are all important parts of maths.

You can chat about these exercises with your friends, family, or anyone else who’s up for it.

1. Write down your own questions.

   As you read this post, note down every question that comes to mind, whether it seems easy or difficult. Some examples for inspiration:

   • In how many different ways can we tile an n × m board with dominoes? Can you describe this number as a function of n and m?
   • What would change if we redefined our pieces to be 3 × 1—three squares long and one square wide? How would the answers to the questions in this post change?
   • What if we chose a different shape, such as an L-shaped piece made from three squares? Which boards could we tile with those pieces?

   Try to guess which of these questions might be easy and which might be difficult. Pick a problem you like, run some small experiments, and make conjectures based on what you find. If you can guess a formula that fits your results, try to prove why it always works.

2. Question a piece of information that seems “obvious.”

   Pick a rule that’s so familiar we accept it without giving it much thought. Here are a few that come to mind, but you can of course choose something else:

   • Why is the product of two negative numbers positive?
   • Why is dividing a number by zero undefined?
   • Why is the zeroth power of a nonzero number equal to 1?
   • Why is the equality $a \times b=b \times a$ true for natural numbers? For example, can you explain why $3+3+3+3=4+4+4$ is true?

   How obvious are these facts, really? How would you explain your chosen rule to a child? If the rule were different, which other properties of maths would it contradict?

3. Try to define a concept.

   Try explaining a concept such as “even number”, “prime number”, “polygon”, “symmetry”, “infinity”, or “randomness” to someone who has never heard of it before.

   If there’s a child around, try explaining it to them. Does your definition include every example it should? Does it accidentally include anything it shouldn’t?

4. Think about what you enjoy about maths.

   As you read this article and work through the exercises above, make a note of anything you particularly enjoy or like discussing with other people.

   What kinds of maths problems do you enjoy thinking about? Have you recently spent a long time on a problem and made real progress—or even solved it completely? How did that process make you feel?

   When you’re not enjoying it, it’s much harder to keep at it consistently. One of the biggest mistakes I made as an undergraduate was not asking myself often enough what I genuinely enjoyed thinking about. So, even though it’s far from an easy question to answer, I think it’s worth pausing every now and then to ask ourselves exactly that.

The domino-tiling questions in this post are adapted from the examples and exercises in the “Chessboard Problems” section of Jay Cummings’s book Proofs: A Long-Form Mathematics Textbook.