Becoming a good chess player requires two separate skills:
Position evaluation.
Strategic search.
Human grandmasters can destroy Stockfish at position evaluation. If you restrict Stockfish to depth one, it’s only around 1000 elo. Maybe 1500. On the other hand, grandmasters are destroyed by Leela at the same task. Yet Stockfish is a better engine than Leela.
This is because Stockfish is much better at search. Most of the elo improvements in the past decade were made by hardcoding in slightly smarter rules for when to prune nodes or when to investigate a little further. The position evaluation has mostly remained the same: a shallow <100k parameter neural network (compare to Leela’s 150MB network).
Getting good at competition math requires two separate skills:
Mathematical knowledge.
Innovation.
Mathematical knowledge does not just include knowing theorems and their proofs, it also includes knowing how to solve specific problems. The knowledge that the Diophantine equation might get simplified by taking modulo seven, because the cubic residues are , and that even if that doesn’t work, maybe something similar will. The knowledge that generating functions kill recurrences and coloring problems, even if the exact generating function takes a few minutes to find.
This is distinct from innovation, the ability to invent new solutions to problem types you have never seen before. You cannot train this skill by learning more math. In fact, that makes it harder to train this skill, because there will be fewer problems you have no clue how to solve. You can only train this skill by bashing your head against a wall, trying a bunch of random things until something sticks. Eventually, you’ll learn what kind of bashing is ‘reasonable’. Until you get hit with a harder problem.
My first experience realizing most people have not trained this skill was in high school. A friend took a math competition for fun, and was explaining their solutions to me. I was flabbergasted. “How could you possibly have come to that conclusion? That’s not a proof, it’s not even a valid heuristic! How did your brain let you write that on the page without stopping to look for the justification?!”
I obviously didn’t say all that. It would have been terribly rude. Also, now that I’m trying to get my younger sister to learn math, I realize they did a lot better than I gave them credit for. They actually tried for several hours. Yes, their trial and error was more flailing than experimentation, but it had hints of enlightenment. There was definitely reasoning going on, and probably reasoning about reasoning too.
I think there are a lot of individuals like this who could get very good at the skill, but never realize it is a skill you can practice. I never set out to practice this skill, I just happened to do math competitions without textbooks or coaches. And it was very frustrating, because I wanted to get better at (1), not (2).
The reason you should not solve other people’s problems for them is not because they don’t want your advice. It is because if you can solve their problem for them, they really need to improve at skill (2). You have less information than them on their specific problem. You are less invested in solving it. They should be better positioned than you to solve this problem. It is not good for their future if you let them offload searching the solution space on you. I sympathize with Confucius. People without the grit to solve their problems should develop that grit, not expect others to subsidize them forever.
The reason you should not solve other people’s problems for them is not because they don’t want your advice.
^Exactly. The main reason I wrote this post was to avoid didactic behavior (ie: lecturing or teaching others too much) and what to replace it with (ie: curious & nonjudgmental questioning).
Becoming a good chess player requires two separate skills:
Position evaluation.
Strategic search.
Human grandmasters can destroy Stockfish at position evaluation. If you restrict Stockfish to depth one, it’s only around 1000 elo. Maybe 1500. On the other hand, grandmasters are destroyed by Leela at the same task. Yet Stockfish is a better engine than Leela.
This is because Stockfish is much better at search. Most of the elo improvements in the past decade were made by hardcoding in slightly smarter rules for when to prune nodes or when to investigate a little further. The position evaluation has mostly remained the same: a shallow <100k parameter neural network (compare to Leela’s 150MB network).
Getting good at competition math requires two separate skills:
Mathematical knowledge.
Innovation.
Mathematical knowledge does not just include knowing theorems and their proofs, it also includes knowing how to solve specific problems. The knowledge that the Diophantine equation might get simplified by taking modulo seven, because the cubic residues are
This is distinct from innovation, the ability to invent new solutions to problem types you have never seen before. You cannot train this skill by learning more math. In fact, that makes it harder to train this skill, because there will be fewer problems you have no clue how to solve. You can only train this skill by bashing your head against a wall, trying a bunch of random things until something sticks. Eventually, you’ll learn what kind of bashing is ‘reasonable’. Until you get hit with a harder problem.
My first experience realizing most people have not trained this skill was in high school. A friend took a math competition for fun, and was explaining their solutions to me. I was flabbergasted. “How could you possibly have come to that conclusion? That’s not a proof, it’s not even a valid heuristic! How did your brain let you write that on the page without stopping to look for the justification?!”
I obviously didn’t say all that. It would have been terribly rude. Also, now that I’m trying to get my younger sister to learn math, I realize they did a lot better than I gave them credit for. They actually tried for several hours. Yes, their trial and error was more flailing than experimentation, but it had hints of enlightenment. There was definitely reasoning going on, and probably reasoning about reasoning too.
I think there are a lot of individuals like this who could get very good at the skill, but never realize it is a skill you can practice. I never set out to practice this skill, I just happened to do math competitions without textbooks or coaches. And it was very frustrating, because I wanted to get better at (1), not (2).
The reason you should not solve other people’s problems for them is not because they don’t want your advice. It is because if you can solve their problem for them, they really need to improve at skill (2). You have less information than them on their specific problem. You are less invested in solving it. They should be better positioned than you to solve this problem. It is not good for their future if you let them offload searching the solution space on you. I sympathize with Confucius. People without the grit to solve their problems should develop that grit, not expect others to subsidize them forever.
^Exactly. The main reason I wrote this post was to avoid didactic behavior (ie: lecturing or teaching others too much) and what to replace it with (ie: curious & nonjudgmental questioning).