Pigeonhole Principle
If n+1 objects go into n boxes, some box has two.
Key idea
If n+1 objects go into n boxes, some box has two.
In depth
If more than n items go into n boxes, some box has at least two. The generalised form: some box has at least ⌈m/n⌉ items. It gives existence results without finding the item.
Worked example
Among 13 people, two share a birth month.
Real-world uses
- Showing that among 13 people two share a birth month
- Hash table collision reasoning
Common mistake
Failing to define the pigeons and holes clearly.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Pigeonhole Principle?
If n+1 objects go into n boxes, some box has two. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Pigeonhole Principle used in real life?
Real-world examples: Showing that among 13 people two share a birth month; Hash table collision reasoning.
What should I learn before Pigeonhole Principle?
Make sure you are comfortable with Cantor’s Diagonal Argument, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Pigeonhole Principle?
Failing to define the pigeons and holes clearly.
How can I practise Pigeonhole Principle?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Pigeonhole Principle?
Move on to Rule of Sum and Product next.
Related advanced topics in Discrete Math & Number Theory
- Tautologies and Contradictions – Always true or always false statements.
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.