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

For $xy-plane$ to be completely covered with regular polygons of $n$ sides, integer number (say $p$) of polygons must meet at a vertex.
Interior angle of a polgon $={180}^{\circ }\left(\frac{n-2}{n}\right)$
So, there must exist an integer $p$ such that $p\times {180}^{\circ }\left(\frac{n-2}{n}\right)={360}^{\circ }$
$\Rightarrow p\left(\frac{n-2}{n}\right)=2$
For $n=3,p=6$
For $n=4,p=4$
For $n=5,p=\frac{10}{3}$
For $n=6,p=3$
Thus, apart from pentagon, rest polygons can cover the plane as stated.