Practice Tiling Intuition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Covering an entire surface with copies of one or more shapes that fit together perfectly with no gaps and no overlaps.

Bathroom tiles cover the floor perfectly—no gaps between them.

Showing a random 20 of 50 problems.

Example 1

challenge
What makes Penrose tilings remarkable compared to ordinary tilings?

Example 2

medium
Does every triangle (not just equilateral) tile the plane?

Example 3

easy
A regular hexagon has 6 sides. How many of them meet at a single corner in a hexagon tiling?

Example 4

medium
Why does the tiling test reduce to checking whether 360°interior angle\frac{360°}{\text{interior angle}} is a whole number?

Example 5

easy
What does it mean to 'tile' a surface?

Example 6

medium
A kitchen floor is 4 m×3 m4\,\text{m}\times 3\,\text{m}. Tiles are 25 cm×25 cm25\,\text{cm}\times 25\,\text{cm}. How many tiles are needed?

Example 7

hard
A hexagon tile is 1 cm1\,\text{cm} on each side, with area about 2.6 cm22.6\,\text{cm}^2. About how many tiles cover a 20 cm×20 cm20\,\text{cm}\times 20\,\text{cm} square?

Example 8

easy
Can regular pentagons tile the plane by themselves (no gaps or overlaps)? Explain using interior angles.

Example 9

challenge
A floor is tiled with squares of side 4 inches. How many tiles cover a 96-inch by 72-inch floor?

Example 10

easy
Which of these regular polygons can tile the plane on their own: equilateral triangle, regular hexagon, regular octagon? Explain using interior angles.

Example 11

medium
Can a regular octagon (interior angle 135°) tile the plane alone?

Example 12

easy
Name the three regular polygons that can tile a plane all by themselves.

Example 13

medium
A regular hexagon has interior angle 120°. How many meet at a tiling vertex?

Example 14

challenge
Why can any quadrilateral (even a non-convex one) tile the plane, while a regular pentagon cannot?

Example 15

challenge
Prove that the only three regular polygons tiling the plane alone are the triangle, square, and hexagon.

Example 16

medium
A regular polygon has interior angle 144°144°. Can it tile the plane alone?

Example 17

medium
Can you mix octagons (corner 135°135°) and squares (corner 90°90°) to tile a floor? Use the corner test.

Example 18

medium
A wall 180 cm×120 cm180\,\text{cm}\times 120\,\text{cm} is covered with rectangular tiles 15 cm×10 cm15\,\text{cm}\times 10\,\text{cm}. How many tiles are needed? Verify using areas.

Example 19

easy
Can squares tile a flat surface by themselves?

Example 20

hard
A regular tile fits at a corner with 44 copies. What is its corner angle?
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