Show that it is impossible to construct an equilateral triangle in the plane whose vertices are lattice points.
My solution was thirteen pages long. It was indeed accepted as one of the correct solutions!
This was not because the problem required thirteen pages. In fact, it only takes a few lines of writing to prove. In contrast, my original submission (PDF) took thirteen pages.
I was in the Polymath Jr. Discord server when Dr. Johanna Franklin posted on the general channel the MAA Metro New York Problem of the Month page. I opened the page while eating my dinner and I immediately started to try to draw some triangles on a plane. I drew some dots on my notepad and pretended that they are all lattice points. After that I drew some right triangles, obtuse triangles, scalene triangles, … and then I realized, ok, what if I just exhaust all possibilities of triangles that can occur on a lattice plane and show that none of them are equilateral? In retrospect, maybe I should have started to think about other possibilities of approaching this question as well, but for some reason that was just the ‘Eureka!’ moment for me. I immediately started to write down all possibilities that can occur. I suspected that the possibilities could be reduced to a small number of broad cases. I needed to make the argument more rigorous.
So I decided to categorize the types of edges that can occur. There are four of them: one vertical, one horizontal, one diagonal with a positive slope, and one diagonal with a negative slope. From there I was able to exhaust all the possibilities purely combinatorially, where there are $\binom{4+3-1}{3} = \binom{6}{3} = 20$ cases, and rule out the theoretically impossible cases (such as three vertical lines or three horizontal lines). I then used the symmetries of the square lattice, described by the dihedral group $D_4$ ,to reduce the twelve remaining cases to five essentially distinct configurations. The rest of the proof eventually utilized some version of the difference angle formula for tangent to show that some rational expression involving integer coordinates would have to equal either $\sqrt{3}$ or $-\sqrt{3}$, which is impossible. Altogether, the submission contained twelve figures, one lemma, four formally stated theorems, and several applications of the tangent subtraction formula.
The actual idea for this proof basically occurred within the first 10 minutes after reading the problem. Writing the actual proof took a lot longer, of course. I wanted to demonstrate each case by a diagram so I spent a lot of time perfecting the diagrams and also showing how the tangent subtraction formula would be used for each case. It took me an entire weekend to write the complete proof as I submitted. You might think it’s not too bad, but I genuinely used up all my weekend, so I think it must have taken like 15-16 hours of pure LaTeX writing and TikZ diagramming to finalize the draft. I guess this is a typical workflow of any mathematical writing: the idea surfaces, you know what to write, but then you need to spend a considerable amount of time to perfect your arguments and make sure your proofs are bulletproof.
As I read through the proof that I wrote a year ago right now, it is comically long and unnecessary. However, it still got to prove the given proposition! I remember last year when I received an email that my solution was accepted as a correct one, I was very proud of my work. Partially it might have been because it was the first substantial piece of mathematical writing I had completed in a while, ever since I graduated college. Even back then, I think the most significant mathematical writing I’ve ever done was a writeup from my Directed Reading Program on category theory, which was a few pages long. However, when I saw the half-page proof on the MAA Metro NY website, I was no longer sure if I should be proud of my work. I thought to myself, ‘Damn, the representative proof fits in half a page???’.
Roughly speaking, Occam’s razor recommends preferring the simpler explanation when competing explanations account for the same facts equally well. I was very much familiar with this idea because this is something that I encountered quite a lot in my Philosophy of Religion course that I took in college. According to that principle, my solution wouldn’t really be the best given that it requires just so much more machinery compared to the short proof which involves basically an application of Pick’s Theorem. I felt bad about myself for a minute, but eventually I came to realize that it was still a worthwhile experience for me because I got to refresh on how to format on LaTeX and also figured out some stuff about how to properly format diagrams on TikZ environments, along with utilizing my abstract algebra knowledge, although it may have been an overkill (as in, this proof could have been written without utilizing orbits). All in all, I had fun trying to come up with an initial idea and formalizing that idea through writing a proof. As Andrew Wiles said, “I loved every minute of it, however hard it had been.” Well, the scale is obviously incomparable, but the basic sentiment still applies: I enjoyed the process of writing this unnecessarily, comically long proof.
As a presentation of this particular result, my proof was obviously inefficient. The half-page proof is easier to understand, easier to verify, and much better for the reader. There was no need for twenty initial cases, five $D_4$-orbits, twelve figures, and repeated applications of the tangent subtraction formula. Still, I do not think writing the longer proof was pointless. It was the approach I saw at the time, and it led to a correct argument. Turning that idea into a complete proof forced me to organize cases, use symmetry, construct diagrams, and make every step precise. More importantly, I enjoyed doing it.
What I understand better now is that the route of discovery and the route of exposition do not have to be the same. An inefficient argument may be how one first reaches a result. Afterward, one can step back and ask whether the reader really needs to follow the entire journey. So no, I do not regret writing thirteen pages. However, for the sake of your sanity, I recommend that you read the shorter proof first. Actually, maybe just leave it at that and don’t read my proof at all 😉
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