All Interesting Problems Are Wrong

Or, what makes a problem interesting

👋 Hey there! My name is Abhishek. Welcome to a new edition of The Sunday Wisdom! This is the best way to learn new things with the least amount of effort.

It’s a collection of weekly explorations and inquiries into many curiosities, such as business, human nature, society, and life’s big questions. My primary goal is to give you some new perspective to think about things.

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Q: Why is it that some problems are widely discussed while others remain largely overlooked?

Let me show you a very low-stakes math problem that you don’t necessarily have to get right. A while back, when it took the internet by storm, it did reveal few aspects about the nature of problems in general. It also shed light on why there are a certain breed of problems that manage to capture our imagination, while others are left ignored (mostly in the hands of “experts”).

When you look at this problem, you realise that each of the numbers on the right is a multiple of the ones on the left. So, there’s a pattern: 9 times 10 is 90. 8 times 9 is 72. 7 times 8 is 56. 6 times 7 is 42. Great! But what is 3?

This question went viral specifically because people were arguing about what 3 is. A lot of people were saying it’s 12. A lot of people were saying it’s 18. Team 18 pointed out that there was a sequence of numbers being multiplied: 10, 9, 8, 7, and 6. So, 6 times 3 is 18.

The 12 people and the 18 people were arguing with each other, and both parties were absolutely sure they were right. Sounds familiar?

It was surprising to see a math problem turn into an Us v Them battle of ideologies, but it was completely pointless because surprisingly the correct answer is: The question is wrong.

The thing about a math problem is that, if it’s ambiguous, it’s a poorly created problem and needs to be rewritten.

But what’s interesting is that people were talking about this particular math problem (instead of every other math problem) precisely because it’s a poorly written problem, and there’s no clear way to arrive at one single objective solution.

There’s after all nothing interesting about a problem (math or not) that everyone agrees upon, thus leaving out any room for intellectual masturbation.

One of the reasons another math problem went viral on the internet was precisely because it was a mess of an equation:

It might seem to an ordinary person (such as you and I) that these are important problems and we need to get to debate them. But mathematicians don’t think that these are half as interesting. Rather than trying to answer such vague questions, they would first ask for these problems to be re-written.

You see, unclear problems are wrong problems to focus on. They mislead us into imagining the world differently, fuelling baseless debates. It’s not just that people believe in different things, instead, when a question is poorly phrased, it becomes much much easy to believe in different things.

For example, after looking at the math problem slightly differently, it is as easy to know that the answer is 12 as it is to know that the answer is 18. Both 12 and 18 are correct answers depending upon how you look at the problem, but good luck convincing the other party of that.

Surprisingly, we spend a lot of our time and energy pondering upon similar poorly phrased questions: Who are better, conservatives or progressives? Should we follow capitalism or socialism? Should there be more government or less government? Should there be open borders or closed borders? Should we be more trustworthy or more sceptical?

Question like these are bound to get a lot of attention, whether they are about math, or economy, government, or pizza. But they mostly draw the attention of people who don’t know much about the topic, because those who know are aware that it’s pointless to spend even an ounce of energy on it.

The irony is that when a question is asked in a more complete way, we’d all agree, and thus we wouldn’t find it interesting enough.

Poorly phrased questions gain undue attention due to the nature of humanity. They’re the ones we see, hear, and argue about the most. And they’re the ones that cause divisions — not just between political factions, but also within a faction or community, and even among people who mostly agree on all problems (but never noticed it because those problems weren’t poorly phrased and it was possible to come to a consensus fairly easily).

Don’t engage with incorrect questions. When you come across one, ask for it to be rectified first.

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I’ll see you next Sunday,
Abhishek 👋