Lesson subsection
Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.
Best flow: read → think on paper → write a short explanation → refine with feedback.
A minimal counterexample is the smallest or simplest counterexample that breaks a conjecture.
Why they matter:
Example:
Conjecture: Every odd integer greater than 1 is prime.
Minimal counterexample: 9 (the smallest odd composite greater than 1).
Another example:
Conjecture: If f'(x) = 0, then f has a maximum at x.
Counterexample: f(x) = x³ has f'(0) = 0 but no maximum.
Minimal counterexamples are especially useful in problem-solving competitions and proof writing.
TL;DR — key idea
Minimal counterexamples isolate exactly where a statement breaks, revealing structural flaws in the conjecture.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
For each false conjecture below, find a counterexample and identify the **smallest** or simplest one: 1. "Every number divisible by 6 is divisible by 12." 2. "Every odd number is of the form 4k + 1." 3. "If n is even, then n/2 is odd." Explain why your counterexample is minimal.
Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.
In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.
AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.