# Tips for writing [proof-golf] questions

As you may know a few days ago I asked you guys to prove (-a) × (-a) = a × a in as few steps as possible and it seems to have been more of a success than I had ever imagined. We've had a lot of fun and generated some really small proofs!

What you may not know is that I asked another question about a year ago. The first question had considerable flaws which I worked to improve for the second, which I think really paid off.

I want to take this knowledge that I have gained and share it with people who are interested in asking about . So what tips do we have for writing questions? This will start as a comparison between the two existing questions, but as we explore the tag we can of course branch out. I know based on vote counts that some users are critical of my second question so I would also love to see what they didn't like about it or what they would do if they were asking a similar question.

Tips should be specific to and should assume that the question writer is already capable of writing a clear, well specified and on-topic question (i.e. tips to improve clarity are not needed). Tips could range from choosing a topic to choosing axioms to choosing a scoring system.

# Build off of people's existing understanding

I think one of the big improvements I made between my first and second question was picking something that most people are already intimately familiar with.

You should try to use mostly concepts that people are already familiar with and build off of their understanding.

For example in (-a) × (-a) = a × a I asked for a substitution style proof. A (less rigorous) kind of this proof is used in most schools as a general problem solving technique and thus is easy to understand.

On the other side of the fence Prove DeMorgan's Laws use a natural deduction style of proof, which is not familiar to most people unless they have studied logic. This can make your question unapproachable for people who might be interested in answering your question otherwise.

This of course goes for things other than choosing a proof system. One should also aim to have the proven result be intuitive.

This has the added benefit of making it really satisfying to solve your problem. I remember when I first learned abstract algebra I loved the feeling of being able to prove concretely things that I had always taken for granted.

• I'm not sure about "aim to have the proven result be intuitive". It might be fun to compete to prove an unintuitive result as concisely as possible... Commented Oct 10, 2017 at 20:36

# Choose the right size of proof

On thing I think is important is that when designing a question you choose a proof that is the right size.

A proof that is too small will only have an optimal answer which people might find in a matter of minutes. A proof that is too large might never get an answer.

Neither of these is optimal so you should keep in mind the size of the proof when writing the questions.

Personally I think if your first solution is around 40 steps that should make a good problem, at least for the abstract algebra style proof.

• When in doubt, I'd say opt for a longer proof instead of a shorter one. I don't think overly long proofs will discourage people for too long, and a long, optimizable proof is better than a shorter, trivial one. Commented Oct 10, 2017 at 1:22
• In particular, you don't want something short enough to be brute-forceable. Commented Oct 11, 2017 at 10:03