Hungarian Problem Books
I used to love maths. For a long time, I wanted to study maths at university. And I did well on tests and in competitions.
Eventually, the plan to study maths at university stopped appealing to me. I was spending my time doing olympiad problems and it was so demoralising to have no clue how to even start. Being in this state for three years while studying maths was not something I could do.
Based on what I hear from people around me, that's how a maths degree can feel at times. I was fortunate to escape.
Shaun spoke to people about the problems of maths education in schools and universities. But he said it wasn't always that way. The Martians had studied maths from problem books. Instead of being told the knowledge taught in schools, they would rediscover it for themselves. While it was harder and produced worse test scores in the short term, in the long term it wasn't just more effective but more enjoyable.
I was intrigued and asked him for a recommendation so that I could try it myself. A few weeks later, I had a book in my hands. After the first few problems, I realised that I had been solving them incorrectly. This wasn't a traditional maths exercise book - I had to apply the process that I'd developed from the workshops to truly appreciate the ideas I was interacting with.
The problems were hard - some took a few days to solve (I spent my evenings working through the problems) and none of them were obvious. But it was always so rewarding to make the discoveries and invent the techniques. The content was not the main focus for me - I had no plans to do research in this area - and it was only a coincidence that what I was discovering turned out to be useful later on. Counting and recurrence relations seem to show up everywhere.
I remember after many hours turning a series of visualisations and diagrams into a recurrence relation for the Bell Numbers and then into an exponential generating function identical to the one I had proved a few days before. And the delight when a visualisation of 7th roots turned into a closed form for a sum. There were many other discoveries and each was innately satisfying to craft.
The quality of my work varied - I was having no input except the standards I held myself to and the solutions in the back (which you are encouraged to look at). When it went well, I could focus for a long time, dancing with the problem and producing beautiful results. When it went badly, it was painful and I made no progress. One afternoon, I did a maths workshop with Shaun. The problem involved creating a decoding algorithm to decompress data streams. While facilitating me, he helped me to overcome some of the bad habits that had been creeping back into my problem-solving process. Later that evening, I proved the existence and uniqueness of a poset's Möbius function by engaging as deeply with a problem from the book as I had done in the workshop. The result was nice but the state I had entered was incredibly rewarding and powerful. It was a magical feeling where all other thoughts evaporated and my full attention focused on my interaction with the problem.
I have no regrets about studying computer science over maths at university but my love of maths was reignited by these problems. I could be a small child once again, playing around in this abstract world.