Logical reasoning, Mathematical proof
The Pigeonhole Principle
Discover how counting objects and categories can prove that a match must happen, even before you know where.
THE KEY IDEA
More objects than places.
If you put more objects into fewer boxes, at least one box must contain two or more objects. The boxes can be real containers, or categories such as colours and days of the week.
Think about the most spread-out arrangement first: one object in each box. Once every box has one, the next object must share a box.
The principle guarantees that a shared category exists. It does not tell us which category it will be, or that every category has a match.
YOUR TURN
Try it yourself
A new challenge: a bag contains red, blue and green counters, with plenty of each colour. You draw counters without looking.
What is the smallest number you must draw to guarantee at least two of the same colour?
Need a hint?
Imagine drawing a different colour each time for as long as possible. What must happen on the next draw?
Show the worked answer
You need 4 counters.
The first three could be one red, one blue and one green, so three do not guarantee a match. The fourth must be one of those three colours, giving at least two of the same colour.
A match could happen sooner, but four guarantees it.
A SHARED TEACHING REFERENCE
Teaching with this video
Use the video as a common starting point, then ask students to explain the structure in their own words.
Discussion prompts & teaching notes
Identify: What are the objects, and what are the boxes or categories?
Explain: Why is three not enough to guarantee a matching pair?
Distinguish: Could happen is different from must happen.
Extend: With four possible colours, five counters guarantee a pair. With three colours, seven counters guarantee three of one colour: six could be spread evenly, two per colour.
Shared vocabulary: category, at least, guarantee, possible, must.