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Thirteen People, Same Birth Month (illustration)
FILE No.64Unsolved

Thirteen People, Same Birth Month

Difficulty ★☆☆about 3 minLogic PuzzlePigeonhole principleWorst case
How many people guarantee that at least two share a birth month?

Case Record

The culprit is someone at a party. The only clue: "the culprit shares a birth month with someone else."

The guest list gives only the number of guests, not their birth months. The detective wants to know how many guests are needed so that someone must share a birth month with another.

There are 12 birth months, January to December.

Your question

What is the smallest number of people that guarantees two or more share a birth month?

Case Files

Evidence

Diagram

The worst casePeopleAll different possible?12Yes (Jan to Dec)13No: two must shareTo be sure, think about the worst case.
Diagram

Timeline

  1. The culprit is among the guests.
  2. Look for shared birth months.

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

Can 12 people all have different birth months?

Open hint 2

What happens when one more person arrives?

Open hint 3

Look for the smallest number that guarantees a match.

Solve It

Your question

What is the smallest number of people that guarantees two or more share a birth month?

Choose the answer closest to the truth.

Share: I'm solving this

The Truth

13

The answer is: 13.

With 12 people, everyone could have a different month (one per month), so 12 does not guarantee a match. With 13 people, at least two must share one of the 12 months. This is called the pigeonhole principle: 13 pigeons in 12 holes means some hole holds two.

12 holes, 13 pigeons12 months = 12 holes13 people = 13 pigeonsSome hole must hold at least two
Diagram of the solution

Key Points

  • For "guaranteed", think of the worst case
  • Worst case (all different) plus one more is the answer
  • Smallest number = months + 1

Why people get fooled

People answer 12, the same as the number of months.

Similar tricks

  • The worst-case type
  • The pigeonhole type

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