Uploaded July 2026 | Updated September 2026, 2 weeks ago
Look at any tiled floor, wall, or footpath and you will see shapes fitting together with no gaps and nothing sticking out. But can every shape be covered this perfectly, or are some shapes impossible to tile no matter how clever you are?
In this lesson you play the tiling game using a single tile that is two squares long. You will see why a rectangle with one even side can always be covered, watch a 5 by 7 board fail with exactly one square left over, and learn to prove impossibility instead of just guessing. We build two powerful ideas: a counting argument on a number line that shows an odd board can never work because an even count can never land on an odd number, and a chessboard coloring trick with blue and purple squares that explains why even an even board, like a 5 by 3 with one square removed, still cannot be tiled when the colors do not match. Then we zoom out to covering the whole plane forever with squares, equilateral triangles, hexagons, and honeycombs, and meet the famous hat tile discovered in 2023 that covers the plane without ever repeating.
Courses on Khan Academy are always 100% free. Start practicing and saving your progress now!
khanacademy.org/math/class-7-ncf/x063f95a40c290960:constructions-and-tilings
Timestamps:
0:00 - Tiling Is All Around Us
0:35 - What Tiling Really Means
1:17 - The Rules of the Game
1:50 - The Easy Case With an Even Side
3:03 - Why the 5 by 7 Board Fails
3:50 - Proving It With Counting
5:06 - When Even Is Not Enough
6:03 - The Chessboard Coloring Trick
7:26 - Tiling the Plane Forever
8:33 - Your Tiling Challenge
Khan Academy India is a nonprofit organization with the mission of providing a free world-class education for anyone anywhere. We have videos and exercises that have been translated into multiple Indian languages and 15 million people around the globe learn on Khan Academy every month.
Support Us: india.khanacademy.org/donate
Look at any tiled floor, wall, or footpath and you will see shapes fitting together with no gaps and nothing sticking out. But can every shape be covered this perfectly, or are some shapes impossible to tile no matter how clever you are?
In this lesson you play the tiling game using a single tile that is two squares long. You will see why a rectangle with one even side can always be covered, watch a 5 by 7 board fail with exactly one square left over, and learn to prove impossibility instead of just guessing. We build two powerful ideas: a counting argument on a number line that shows an odd board can never work because an even count can never land on an odd number, and a chessboard coloring trick with blue and purple squares that explains why even an even board, like a 5 by 3 with one square removed, still cannot be tiled when the colors do not match. Then we zoom out to covering the whole plane forever with squares, equilateral triangles, hexagons, and honeycombs, and meet the famous hat tile discovered in 2023 that covers the plane without ever repeating.
Courses on Khan Academy are always 100% free. Start practicing and saving your progress now!
khanacademy.org/math/class-7-ncf/x063f95a40c290960:constructions-and-tilings
Timestamps:
0:00 - Tiling Is All Around Us
0:35 - What Tiling Really Means
1:17 - The Rules of the Game
1:50 - The Easy Case With an Even Side
3:03 - Why the 5 by 7 Board Fails
3:50 - Proving It With Counting
5:06 - When Even Is Not Enough
6:03 - The Chessboard Coloring Trick
7:26 - Tiling the Plane Forever
8:33 - Your Tiling Challenge
Khan Academy India is a nonprofit organization with the mission of providing a free world-class education for anyone anywhere. We have videos and exercises that have been translated into multiple Indian languages and 15 million people around the globe learn on Khan Academy every month.
Support Us: india.khanacademy.org/donate
