Getting Unstuck on a Maths Problem: A Patient Method

Last winter I spent most of a tram ride to the city glaring at one sentence in a puzzle book. It was a simple-looking question about people shaking hands at a party, and I couldn't get a grip on it. I read it four times. I got off at my stop feeling faintly foolish, as if everyone else had been handed the trick at the door.

The answer, when I got there, came from doing something humbler than thinking harder. I made the problem smaller. A neighbour who used to teach had once told me, over a fence and a lemon tree, that "stuck" usually means the problem is too big to hold. Her advice was to stop trying to hold it.

So here is the short answer to the question in the title. When a maths problem won't move, don't push harder in the same direction. Restate it in your own words, try a smaller version, draw something, and then step away for a while. Each of these does a different job, and I'll explain what each one is for. This is what has worked for me. I'm still learning it, and it doesn't work every time.

Why does a maths problem make you feel so stuck?

Being stuck feels like a verdict, but it's closer to a weather report. If a problem were easy, you'd already be finished. The fact that you're sitting there not knowing what to do tells you the problem is a real one, not that you are a poor mathematician.

My library's returns desk taught me something about this, oddly enough. Most of the work there is waiting: for the trolley to fill, for the shelf to make sense, for the one misfiled book to turn up. Nothing is wrong when the book isn't there yet. The work simply hasn't finished. Hard problems have the same shape. There is a long middle where nothing seems to happen, and that middle is part of the job, not a failure of it.

Here is the problem I'll use to show the method:

Ten people are at a party. Every person shakes hands with every other person exactly once. How many handshakes happen in total?

It's a small problem, but it has the right feel. You can start in a dozen directions and none of them obviously works.

Step one: say the problem in your own words

Before calculating anything, close the book and say the problem aloud, or write it down as if explaining it to a friend. Don't copy the wording. Use your own.

For the handshakes, my first restatement was clumsy: "Everyone meets everyone, once." My second was better: "I'm counting pairs of people. Each pair shakes hands one time." That small shift mattered. I stopped thinking about handshakes, which are events, and started thinking about pairs, which are things I can count.

Why does this help? Written problems are compressed, and the compression often hides the structure. Putting it in your own words forces you to decompress it, and you find out quickly whether you understood it at all. Sometimes the discovery is that you'd misread a detail. In this case, it might be whether a handshake between A and B counts separately from one between B and A. (It doesn't. One handshake involves two people.) Catching that on step one saves a lot of confused arithmetic later.

It's also a gentle step. It asks nothing of your cleverness, only your attention.

Step two: try a smaller version

Descartes, in his Discourse on Method of 1637, gave advice that I'd paraphrase like this: when a difficulty is too large, divide it into as many parts as you need and work through them in order, beginning with the simplest. It's four centuries old and still fits a Tuesday afternoon.

For the handshakes, the smaller version is to shrink the party. Let's count by hand:

• 2 people: 1 handshake
• 3 people (call them A, B, C): AB, AC, BC, so 3 handshakes
• 4 people (A, B, C, D): AB, AC, AD, BC, BD, CD, so 6 handshakes
• 5 people: 10 handshakes

Look at the totals: 1, 3, 6, 10. The gaps between them are 2, 3, 4. A pattern is peeking out.

Why does a small case reveal this? With ten people, there are too many handshakes to hold in your head, so the structure is buried under the detail. With three people, you can see every handshake at once, and the structure has nowhere to hide. A simple case is not a cheat or a retreat. It's a way of looking at the same problem under better light.

And the pattern has a reason. Imagine the party grows by one person. The newcomer shakes hands with everyone already there. That adds as many handshakes as there were people before. So going from 4 people to 5 adds 4 handshakes, which takes 6 to 10. Going from 9 people to 10 adds 9. The total for ten people is therefore:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.

A warning about patterns

Here I need to slow down, because there's a common misconception about this step. It's tempting to believe that once you see a pattern in small cases, you've found the answer. You haven't, quite. You've found a good guess.

There's a well-known example of how a pattern can mislead. Put some points on a circle and join every pair with a straight line. Count the regions inside the circle. With 1, 2, 3, 4 and 5 points you get 1, 2, 4, 8 and 16 regions, which looks exactly like doubling each time. With 6 points, you'd expect 32, but the count is 31. The pattern was a coincidence for a while, and then it stopped.

That is why, in the handshake problem, I didn't stop at "the gaps go 2, 3, 4." I asked why they go that way, and found the reason: each newcomer shakes hands with everyone already present. The pattern gives you a guess, and the reason gives you something you can trust. Small cases are for finding the question worth asking, and the "why" is for checking it.

Step three: draw something

Drawing may feel like a childish step, but it does real work. Words and numbers are good at sequences, and pictures are good at relationships. Many problems are really about relationships, and a sketch lets your eyes do part of the thinking.

For the handshakes, draw ten dots in a circle, one for each person, and draw a line between every pair. You'll get a tangle, which is satisfying in its own way. Now count the lines from just one dot. Each person has lines to the other nine. That gives 10 × 9 = 90 "line-ends" in total.

But every line has two ends, one at each person, so I've counted every handshake twice. Halve it: 90 ÷ 2 = 45. This matches the 45 from the pattern, reached by a completely different route. When two routes agree, I trust the answer more than I would trust either alone.

It also gives a general rule for any number of people, n: each person shakes hands with n − 1 others, every handshake is counted twice, so the total is n × (n − 1) ÷ 2. Check it against the small cases: for 4 people, 4 × 3 ÷ 2 = 6. For 5, 5 × 4 ÷ 2 = 10. Both match what we counted by hand.

I'd add a small honest note. I didn't see this in the drawing straight away. My first sketch was a mess, and I miscounted twice before I thought to count from a single dot. Drawing is allowed to be untidy and to take a few goes.

Step four: walk away for a bit

The last step is the one that feels like cheating and often isn't. When you've restated, shrunk and drawn, and you're still stuck, put the problem down and do something else.

The mathematician Henri Poincaré, writing in the early 1900s, described a moment like this. He had been working on a difficult problem without success, set it aside, and went on a trip. As he stepped onto an omnibus, an idea arrived that he hadn't been pursuing at that moment. He took it as a sign that his mind had been working on the problem while he wasn't aware of it. I'm paraphrasing from memory of his account, and he was telling a story about his own experience, so I wouldn't treat it as a law of nature.

What I can say plainly is what I've seen in my own days. Stepping away does two sensible things. It lets you drop a wrong approach that you've been gripping out of stubbornness. And it lets you come back with a fresh read, much as you'll notice a typo in your own writing the next morning that you couldn't see the night before. Does the "background thinking" idea hold up? Researchers still discuss how that works, and I'm not able to settle it. Even so, a walk around the block, the washing-up, or a cup of tea costs nothing, and I've never regretted trying it.

Two cautions: walking away is not the same as giving up, and it works best after you've made a real attempt. Rest helps when your mind has something to chew on. A problem you've only glanced at gives it nothing to work with.

What if the four steps still don't work?

Sometimes they won't. I've left problems overnight and returned to find them as stubborn as before. A few further things I try, none of them magic:

• Check the problem is the one you think it is. Re-read it for a word you skipped.
• Try an even smaller case. If two people feels too trivial, that's usually a sign it's the right size.
• Write down what you do know. Even a small, obvious fact can be a foothold.
• Ask someone. A neighbour, a teacher, a friend. Explaining where you're stuck often shows you the gap before they say a word.

None of these promise success on a given day. What they offer is a way to keep making small progress instead of sitting in the fog.

Being stuck is the middle, not the verdict

If I could pass on one thing from all this, it's that being stuck is not evidence against you. It's how the middle of hard work feels for nearly everyone, including people who do mathematics for a living. The difference isn't that they get stuck less. It's that they have a few steady things to do while they are.

So the next time a problem refuses to move, you might say it again in your own words, count a tiny version by hand, scribble a picture, and go and put the kettle on. Small steps count, and a half-understood pattern or a drawing with a mistake in it is still progress. Thank you for reading, and I hope the method is a kind companion on the next hard one.

A question to wonder about

The handshake total, 1 + 2 + 3 + … + 9, is the same as the number of dots in a triangle of rows, one dot in the first row, two in the second, and so on. It also turns up when counting pairs in a committee, games in a round-robin tournament, and diagonals in certain shapes. Why should the same quiet pattern appear in so many unrelated places? I have a partial answer, which is that all of them are secretly about choosing two things from a group. But I'm not sure that explains the whole of the pleasure, and I'd like to know what you think when you next spot it.