Hobby maths in the age of AI

Over the years, I have accrued a series of puzzles that I either solved myself or were given to me, whose solutions I find interesting or beautiful enough to hold on to. The first one came from a problem I solved in a day, that I generalised as far as my solution would go. The second was similar, but it took me a course in group theory to fully nail down a generalised solution. The third puzzle was given to me by a lecturer who took a year to solve it. I worked on it in a group and we eventually cracked it after several months of collective effort, but it is certainly the most difficult and interesting puzzle that I have solved outside of a literal research problem. (I’ve provided these problems at the end of this post.)

I like to use these puzzles as a benchmark to test new LLMs. Partly because it gives me some comfort to see the AIs struggle miserably, flailing around with useless thoughts, failing to comprehend the salient aspects of the problem even with the strongest hints.

Until they start solving the problems. The second one, first to fall, was claimed by Opus 4.6 in February. The first was solved by Gemini five days later. And finally, on the 26th of July 2026, GPT 5.6 Sol solved my last problem, without any hints, after 15 minutes of thinking.

I am an obsessive programmer. In 2021, I wrote a web app to visualise some of my research. I spent almost a week on a first pass, followed by multiple blocks of days where I did nothing but code to really pin everything down. I’m talking 16-hour days for 3 days straight. During one of the busiest times of my life, with a dissertation deadline looming. Laptop in bed, not eating enough, not sleeping enough, withering away until the vice grip of unrealised potential finally released me. Fortunately it turned into a dissertation (which I mostly wrote over the course of 3 weeks), but it was not healthy or a great use of my time.

Two years later I wrote another web app for my next research, following the same approach. Then I rewrote them in another framework. Then some abandoned visualisers, and some art, and a website, and a blog, and another blog. Every time wondering whether the effort was worth it. Sometimes I wonder what I’m even doing it for.

I don’t even like coding that much. I’m not motivated by the process. What really motivates me is a desire for the destination. There is a void in the world, caused by an inability to write down a pair of matrices and instantly see how their Cayley graph embeds into a Poincaré ball, that grates at me until it is filled. Then I invariably seek out the next void, and begin again.

What am I reaching for? From a concrete perspective, it’s a program that does something meaningful and looks cool and is fun to play with, and succesfully uses my new favourite web framework. From an abstract perspective, it’s information. When I look at examples, when I visualise or transform a problem in a way that exposes its hidden structure, I feel a satisfaction at the raw information available at my fingertips. Secrets laid out in front of me, free for the taking like the sweets in a well-stocked conference breakroom.

(I’m lying a little about not enjoying writing code. Code itself can have those little glimmers of satisfaction, when it lines up just right and reveals that hidden structure.)

Nowadays, I just prompt Opus with what I want and it makes the visualiser for me. Tweak the CSS, shuffle some things around, ask it to add this or that feature, and out comes the app of my dreams—its interface dynamic, seamless and flawless. Applying just as heavy a hand as I need to make the output reflect my inner world.

The interesting thing is this: when the LLMs code up programs in hours that would have taken me days by hand; when they output solutions to my puzzles, more elegant and efficient than I or others have come up with on our own; I feel that same satisfaction of witnessing something beautiful.

I’m not all that motivated to solve puzzles anymore, now that I can just chuck them into Claude, Gemini, or GPT for an answer. Honestly I’m fine with that. I like learning the solution more then solving it anyway, and now I get to do more of that.

And it means that I can spend time working on harder things, that I would not have been able to otherwise. In my previous post, it was Opus that clued me into the connection between the Hookean-spring-based field potential and the polar moment of inertia. It was also Opus that suggested I look at the medial axis, which led to the grassfire transform. I’ve been using LLMs to connect questions I have to fields that I didn’t know anything about, and because of it I’m able to learn so much more.

It’s tempting to imagine what I would do as a research mathematician. Imagine all of the examples I wouldn’t have to work out by hand! All of the lemmas I wouldn’t have to prove, the counterexamples I wouldn’t have to find, while I explore my way to the bigger ideas that can’t be prompted! And if the AI fails, well, that just proves that they still need me after all.

There are a lot of mathematicians who are scared for their field and their future. They are afraid that everything they value about maths is going to become obsolete. The jobs will either be lost to AI, or maths will become about prompting a machine to do all of the would-be meaningful work for you.

This could all be true. I don’t have any more answers than anyone else. I have no idea what will happen to mathematics. I might not be the best person to comment, as someone who has never experienced the divine in maths, or anything else really. All I know is that to me, right now, it feels like an upward trajectory, a tool that can launch you to new heights.

But then, I’m not an academic anymore and I have no stake in the game. My only worry is trying to find a job, when everyone is buying tokens instead of hiring juniors and the candidate pool is too competitive for a lowly pure maths PhD graduate with almost no industry experience. Maybe it’s a sign of things to come, and everybody should be worried. Maybe all of this is small potatoes, and we’ll laugh at how we fussed about jobs and meaning while the nanobots disconnect our brain stems.

I don’t think I have it in me to be worried about the future. All I can do is focus on the present, doing what I can to try to get a foothold in life and find joy where I can get it, whatever ‘joy’ is. And right now, having an assistant that can guide me through mathematical rabbit holes can be pretty neat sometimes, maybe even joyful.

… I really need to get out of the house.

Addendum

Here’s my list of puzzles. The second one requires graduate-level maths knowledge, the other ones technically don’t but it would certainly help.

1. Prisoners with prime power hats

Let q be a prime power. There are q+1 prisoners in a prison. The warden positions them so that each prisoner can see all but one other prisoner, and can be seen by all but one other prisoner. Each prisoner does not know who the others can see, but they can recognise the prisoners they do see.

The warden places a hat on each prisoner’s head. Each hat is one of q different colours (repeats are allowed). The prisoners cannot see their own hat. On the warden’s command, the prisoners must simultaneously guess the colour of their own hat. If one prisoner guesses correctly, everyone is let free. Otherwise all prisoners are killed. The prisoners may strategise beforehand, but once arranged they cannot communicate. How can they guarantee freedom?

2. Generalised blind bartender

Alice and Bob play the following game. First they agree on a finite group $G$ and a finite abelian group $V$ on which $G$ acts faithfully. Bob secretly chooses some $x_0 \in V$. On the $i$-th turn, Alice tells Bob some $y_i \in V$, Bob chooses some $g_i \in G$, then Bob assigns $x_i = g_i (x_{i-1}+y_i)$. If $x_i = 0$ for some $i$, then Alice wins. Characterise $G$ and $V$ such that Alice has a guaranteed winning strategy.

3. Prisoners in a ring

You and an unknown number of prisoners are placed in separate jail cells. The cells are connected in a ring, so that each cell has one cell to its left and one to its right. Every cell is occupied. Every day, you and each prisoner must send a bit signal (1 or 0) to the cell on their right, which is received at the end of the day. After the signals are received, the warden rearranges the prisoners. At no point do you or the other prisoners know what cell they are in.

The goal is to determine the number of prisoners; on any day you may guess the number of prisoners, and if you are right then everyone is freed. If you are wrong then everyone is executed, so you only get one guess.

You may communicate a strategy to the other prisoners beforehand, but they all receive the same strategy and there is no ability to distinguish them.

Find a (deterministic) strategy that guarantees freedom.