COSMO · STUDY + TEACHING DECK

Problem Solving

June 21, 2026

A question worth sharing

Invariants

Track what cannot change, and impossibility becomes visible.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use state, move, invariant 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

A standard chessboard loses two opposite corners. Can the remaining board be tiled by 1 x 2 dominoes?

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

Color the board as a normal checkerboard.
Count removed black and white squares.
Ask what every domino always covers.

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.

state

the current configuration

move

an allowed change

invariant

a quantity or property every move preserves

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

The core idea

Keep the idea. Explain the conditions.

I(S_{t+1})=I(S_t)\quad\Longrightarrow\quad I(S_t)=I(S_0)

The idea

Track what cannot change, and impossibility becomes visible.

What to remember

Do not follow every move; track the information every move cannot erase.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Why can the mutilated chessboard not be tiled?

Predict each next step

Opposite corners have the same color.
Removing them leaves 30 of one color and 32 of the other.
Every domino covers one square of each color.
A complete tiling would require equal counts, so it is impossible.

Which step needs a reason, not just a calculation?

See the structure

Coloring exposes information shape alone hides

Removing opposite corners leaves unequal color counts.

Read the representation

The checkerboard coloring is not decoration. It creates a count that every legal domino placement preserves.

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.

Invariant

Exactly preserved by every move.
Useful for impossibility.
Examples: parity, color balance, modular class.

Monovariant

Moves only one direction.
Useful for termination.
Examples: energy decreases or disorder count drops.

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

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

Moves only one direction.
Useful for termination.
Examples: energy decreases or disorder count drops.

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

Invent a useful coloring

Your shared task

Groups try two-color and three-color schemes on a board puzzle, then defend which one reveals the obstruction.

Explore

5 min - test colorings

Compare

5 min - track one move

Defend

4 min - state the invariant

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

An invariant must be checked against every allowed move.
The starting and target states can be compared without simulating every path.
A failed invariant may still suggest a stronger modulus or coloring.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Starting at 0, each move adds 4 or subtracts 6. Can you reach 101?

Predict each next step

Both moves change the number by an even amount.
Parity therefore stays even.
The starting value 0 is even.
101 is odd, so it is unreachable.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A strictly decreasing nonnegative integer-valued quantity proves a process must stop.

Build the argument

Choose an integer-valued quantity that drops on every move.
It can never fall below zero.
Only finitely many decreases are possible, so the process terminates.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

For a coin-turning game, decide whether parity of heads is preserved, flipped, or useless under each allowed move.

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

Board puzzles, algorithms, games, parity arguments, and termination proofs become manageable when one stable quantity is found.

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

What must be checked before calling something invariant?

Question 2

Why does the chessboard coloring prove impossibility?

Question 3

How is a monovariant different?

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

Do not follow every move; track the information every move cannot erase.

On the worksheet, write START, MOVE EFFECT, and TARGET before attempting a simulation.

A good question belongs to everyone in the room.