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Strategy 04 of 12Backward Thinking
Start from the goal — work backwards
Goal-OrientedConstructionInequalities
Like solving a maze from the exit! Instead of starting from the data and trying to reach the conclusion, start from the conclusion and ask: 'What must be true for this to hold?' Often the goal has more structure than the hypotheses.
When to use it
- →When the goal is more specific than the data
- →When you don't know what to do with the hypotheses
- →When the goal has special form (equality, divisibility)
- →In construction problems
How to think (step by step)
- 1Write the goal: What exactly must you prove?
- 2Ask: 'What would imply this?' — What condition suffices?
- 3Work backwards: At each step, ask 'What do I need for this?'
- 4Connect to the data: When you reach something known, stop
- 5Reverse: Write the proof from hypotheses to conclusion
Practice Problems
Three problems at increasing difficulty — try each before revealing the hint or solution.
📘Basic
Given positive numbers \(a, b, c\) with \(a + b + c = 1\). Prove that \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9\).
📙Intermediate
Construct a segment of length \(\sqrt{5}\) using only compass and straightedge.
📕Advanced
Let \(a, b, c > 0\). Prove: \((a+b)(b+c)(c+a) \geq 8abc\).