Playground & Quiz

Practice problems and test your proof technique knowledge

Playground & Quiz

Experiment freely with the Pigeonhole Principle playground, then test your understanding of all proof techniques with a comprehensive quiz.

Pigeonhole Playground

Experiment with different numbers of pigeons and holes. Build intuition before taking the quiz!

Basic Pigeonhole Demo

Drag the pigeons into the holes. When you have more pigeons than holes, at least one hole must contain 2+ pigeons!

5
4
You have 5 pigeons and 4 holes. Since 5 > 4, the pigeonhole principle guarantees at least one hole will have 2+ pigeons. Try placing all pigeons!

Key Insight

If n+1 objects are placed into n containers, at least one container must hold more than one object. This simple fact is surprisingly powerful in mathematics!

Quiz: Test Your Knowledge

30 questions covering all proof techniques. See how well you understand the concepts!

Proof Techniques Quiz

Score: 0/0
Question 1 of 30

Topic: Basic Principle

If you have 5 pigeons and 4 pigeonholes, what does the Pigeonhole Principle guarantee?

What You've Learned

  • Direct proof -- assume P, derive Q step by step
  • Pigeonhole Principle -- more objects than containers guarantees a collision
  • Mathematical induction -- base case + inductive step proves all natural numbers
  • Well-Ordering Principle -- minimum counterexample technique
  • Contradiction -- assume ¬P, derive impossibility
  • Contrapositive -- prove ¬Q → ¬P instead of P → Q