From: 3blue1brown
Overview
Problem-solving strategies in mathematics, particularly for challenging contest problems, often involve an approach that emphasizes exploration, the formalization of ideas, and the search for invariants [14:12]. These strategies are crucial for tackling “pure puzzles” that do not necessarily rely on specific theorems but rather on finding a “clever perspective” [03:39].
Characteristics of Challenging Problems
Contest math problems, such as those found in the International Math Olympiad (IMO), are typically proofs that require discovering and articulating a rigorous line of reasoning [00:34]. Some problems, like IMO 2011’s Problem 2 (the “windmill puzzle”), stand out as unusually pure puzzles [03:25]. These types of problems do not test knowledge of a particular theorem, but rather the ability to find a clever perspective [03:39]. Despite their apparent simplicity in terms of prerequisite knowledge, they can be extremely difficult even for world-class problem solvers [03:55], [05:04], demonstrating that “knowing when math is hard is way harder than the math itself” [14:44].
Key Problem-Solving Approaches
1. Playing Around and Simplification
The first step with any puzzle is to “simply play around with it and get a feel for it” [05:18]. It is always beneficial to start with the simplest case and gradually increase complexity [05:23]. This involves:
- Drawing Examples: It is often helpful to start drawing examples to visualize the problem as it is being read [01:35], [02:05].
- Physical Simulation: For geometric problems, physically moving a pencil among drawn dots can aid understanding [06:18].
- Belief in the Result: It is important to “believe a result before you try too hard to prove it” [06:21].
2. Formalizing Vague Ideas and Quantification
When a vague idea feels productive, it should be made more exact, preferably by quantifying it [07:39], [07:47]. For instance, in the windmill problem, the vague idea of a line being “in the middle” was formalized by counting the number of points on either side of the line [07:53].
3. Seeking Invariants
A fundamental mathematical approach to solving complex systems is to find something that stays constant during the unfolding process [14:12], known as an invariant [14:24]. This concept is ubiquitous in mathematics and physics, from counting holes in a surface (topology) to defining energy and momentum (physics) [14:27]. Identifying an invariant can lead to a “slick solution” [15:26].
Example: The Windmill Problem (IMO 2011, Problem 2)
This problem highlights these strategies:
- The Problem: Given a finite set of at least two points on a plane, with no three points collinear, a “windmill” process involves a line rotating clockwise around a pivot point until it meets another point, which then becomes the new pivot [01:51]. The question asks to show that a starting point and line can be chosen such that the windmill uses each point of the set as a pivot infinitely many times [02:42].
- Initial Exploration: Playing with 2, 3, and 4 points reveals that some starting conditions lead to all points being hit, while others do not [05:27]. The idea that a line starting “in the middle” tends to stay there emerges [06:58].
- Identifying the Invariant: The key insight involves counting the number of points on either side of the line [07:53]. By observing how the line changes pivots, it can be proven that the number of points on a given side of the line remains constant throughout the process [09:54], [10:07].
- Solution: This invariant proves that if the line starts with an equal number of points on either side (or a specific configuration for an even number of points), after a 180-degree rotation (or 360 degrees for an even number), the line returns to a state where it has hit all points, and the cycle repeats indefinitely [11:07], [12:23].
Broader Implications for Mathematical Education
Puzzles and mathematical problems are comparable to fables in understanding real life; they carry lessons relevant to useful, real-world problems [15:01], [15:13]. Solving problems that require finding an invariant helps students appreciate that fundamental mathematical definitions were once clever discoveries themselves [14:42], [14:46].