COSMO · STUDY + TEACHING DECK

Counting

September 25, 2025

A question worth sharing

Pigeonhole Principle

More objects than containers force at least one collision.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use pigeon, hole, ceiling to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

Why must two of 13 people share a birth month?

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

pigeon

object being assigned

hole

category receiving objects

ceiling

rounding up a forced average

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

\max_i n_i\ge\left\lceil\frac{N}{k}\right\rceil,\qquad\sum_{i=1}^{k}n_i=N

The idea

More objects than containers force at least one collision.

What to remember

A forced collision becomes obvious once the right containers are named.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Show that among six integers, two have the same remainder mod 5.

Predict each next step

The holes are remainders 0,1,2,3,4.
There are five holes.
Place six integers by remainder.
One remainder class receives at least two.

Which step needs a reason, not just a calculation?

See the structure

The structure behind pigeonhole principle

\max_i n_i\ge\left\lceil\frac{N}{k}\right\rceil,\qquad\sum_{i=1}^{k}n_i=N

When N objects enter k boxes, with k at least 1, one box reaches the rounded-up average.

Read the representation

More objects than containers force at least one collision.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate pigeonhole principle

Your shared task

Students design their own forced-collision claim by choosing the objects and holes explicitly.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

A forced collision becomes obvious once the right containers are named.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

How many students force at least four to share a birth month?

Predict each next step

At most three may occupy each month without reaching four.
Twelve months then hold at most 36 students.
One more student forces a fourth.
The answer is 37.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

Placing N objects into k boxes forces some box to hold at least ceiling(N/k) objects.

Build the argument

If every box held at most ceiling(N/k) - 1, the total would be too small.
Sum the assumed upper bounds across k boxes.
The contradiction proves the forced occupancy.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Choose 11 numbers from 1 through 20. Prove that two are consecutive.

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

The principle proves duplicate hashes, repeated remainders, nearby points, and unavoidable coincidences.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

Why must two of 13 people share a birth month?

Question 2

Explain why this claim is true: Placing N objects into k boxes forces some box to hold at least ceiling(N/k) objects.

Question 3

Choose 11 numbers from 1 through 20. Prove that two are consecutive.

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

A forced collision becomes obvious once the right containers are named.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.