Question

Suppose we have to cover the $xy-$plane with identical tiles such that no two tiles overlap and no gap is left between the tiles. Suppose that we can choose tiles of the following shapes; equilateral triangle, square, regular pentagon, regular hexagon. Then the tiling can be done with tiles of

• A. all four shapes
• B. exactly three of the four shapes
• C. exactly two of the four shapes
• D. exactly one of the four shapes

Answer 1

The correct option is B exactly three of the four shapes

We can cover the plane using squares definitely. Using equilateral triangle, we can also cover the plane. Also regular hexagon is made of equilateral triangles. But pentagon cannot cover the plane because of its shape.