Invariants · Explore together
What can a domino change?
Remove squares, try covering the board, and look for a quantity every placement preserves.
One person controls the experiment. The other predicts and questions. Swap roles after a trial.
Can this board be tiled? Each domino covers two side-by-side squares. Try placing a few, then look for information that every placement preserves.
Select a free square, then a free neighbor sharing a side.
Select two free adjacent squares to place a domino. Click a placed domino to remove it. In square-edit mode, a square can be removed or restored.
A question to discuss
What happens to the difference between the two uncovered color counts after one placement? Does equal color balance alone guarantee a tiling? Try the balanced mystery.
Explain without finishing. Use the starting board, the effect of every legal placement, and the desired final board to justify a conclusion. A failed attempt alone does not prove impossibility.
Compare your evidence
What can you explain now?
Ask your partner to restate your reasoning. An experiment suggests a pattern; your explanation should say why it holds and when it applies.
- What color balance does each domino preserve?
- Why can failed attempts alone not prove impossibility?
- Do equal color counts guarantee a tiling, or only pass one necessary check?