← All Strategies
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Strategy 09 of 12Reformulation
Say the same thing in a different way
SubstitutionTranslationEquivalent Problems
Transform a problem into an equivalent one that is easier to solve. Change variables, translate to a different domain, or reinterpret the question. The key insight: if two problems are equivalent, solving either one solves both.
When to use it
- →When the problem looks complex in its current form
- →When there is hidden structure
- →When you know a similar solved problem
- →When a change of language can simplify
How to think (step by step)
- 1Understand the problem: What exactly is asked?
- 2Identify the obstacle: What makes it hard?
- 3Think of alternatives: How else could I state this?
- 4Apply the transformation
- 5Solve the new problem
- 6Translate back to the original
Practice Problems
Three problems at increasing difficulty — try each before revealing the hint or solution.
📘Basic
Solve: \(\sqrt{x + \sqrt{x + \sqrt{x + \cdots}}} = 2\)
📙Intermediate
Prove: for positive \(a, b, c\) with \(abc = 1\), \(\frac{1}{1+a+b} + \frac{1}{1+b+c} + \frac{1}{1+c+a} \leq 1\).
📕Advanced
Prove: in any \(2\)-coloring of the positive integers, there exist \(a, b, c\) of the same color with \(a + b = c\).