How to think like a genius ( from a 5x IMO medalist)

Summary

The interview focuses on how strong mathematical problem-solvers approach unfamiliar problems. Its central lesson is to look beyond memorized solutions and analyze the underlying ideas that may apply across many problems.

Main ideas and methods

• Analyze problems deeply rather than memorizing them. A problem’s exact statement is unlikely to recur. Instead of remembering only its solution, identify the general ideas and structures that made the solution work.
• Build a mental map of patterns and “triggers.” Certain features in a problem may suggest useful methods:
  • Expressions involving powers of 2 may suggest induction, especially sequences such as (2, 4, 8, 16, \ldots).
  • An equation involving squares, such as (x^2 + y^2 = 3), may prompt checking modular arithmetic, particularly modulo 4 or 8.
  • Other problems may offer less explicit cues toward tools such as the pigeonhole principle.
• Learn to recognize what suggests a method. Knowing that a technique can solve a problem is not enough. When practicing, ask what feature of the problem pointed toward that technique. This makes the connection easier to recognize in future problems.
• Treat the mental map as a guide, not a rulebook. Mathematical cues indicate promising approaches, but they are not guarantees. A map based on experience can be refined, but it can also make an unusual route seem too unlikely to consider. Some problems are designed to reward an unexpected approach.
• Stay willing to change direction. The interviewee describes spending about three and a half hours on IMO Problem 5. He first pursued what seemed like the most probable approach, including ideas such as binary search. When that route did not work, he tried another direction that could lead to a solution he could write up quickly. He found a solution, while acknowledging that luck played a part.

The interviewee’s interpretation of the problem’s difficulty

The interviewee suggests that an unusually low-probability approach may help explain why experienced contestants, including the Chinese team, did not solve that year’s Problem 5, while some less-experienced students found it more readily. He presents this as a possible explanation, not a certainty.

Speakers and sources

• Interviewer: Asks about problem-solving habits, mathematical cues, and the interviewee’s experience with IMO Problem 5.
• Interviewee: A Malaysian five-time IMO medalist with two gold medals who explains his approach to recognizing patterns and recounts his experience solving Problem 5.
• Introductory narrator: Introduces the interviewee and frames the discussion.