Korva is a scroll of the internet’s best ideas.
The troll problem that beat Terry Tao
People called it a troll problem. Everything about the context, an International Math Olympiad question and the way contest math is trained, pointed toward a really elegant route that feels like the solution. That route is enticing, yet it is hard to prove it is the best, and the reason is that it is not. The best answer is an almost brain-dead one. Someone handed the problem as a brain teaser on the street would probably answer it well.
Solving it takes escaping the context. Human researchers run into the same thing, and there are stories of people who worked on a proof for a long time before asking what would happen if they tried to prove the opposite. A person cannot wipe their own context on demand. A system of AI agents could be built so that each starts with different pieces of context, one trying to prove a statement, one trying to disprove it, and one attacking it a different way.
Why does a first method block simpler ones?
In 1942, psychologist Abraham Luchins gave people water jar puzzles. After they learned a long method for several problems, many kept using it even when a much shorter solution was available. He called this the Einstellung effect. Groups who had not learned the long method found the short one easily.
How are Olympiad problems chosen?
Each participating country submits proposed problems, and an organizing committee narrows them to a shortlist. The jury of team leaders then picks six, with the aim of a range from easy to very hard. Students sit three problems on each of two days, with four and a half hours each day. Terry Tao took part at ten, eleven, and twelve, and won gold at thirteen in 1988.
How does disproving help prove something?
A proof by contradiction assumes the statement is false and derives an impossibility. Searching for a counterexample is the other side of the same coin: if one exists, the statement is false, and if every attempt breaks down, the reason it breaks often points to the proof. Many famous results were found this way, including the irrationality of the square root of two, which dates to ancient Greece.
More from Dwarkesh Podcast
