Problems

Irreducibly tile a floor with \(1\times2\) tiles in a room that is a \(6\times8\) rectangle.

In the \(6\times7\) large rectangle shown below, how many rectangles are there in total formed by grid lines?

Prove that one can tile the whole plane without spaces and overlaps, using any non self-intersecting quadrilaterals of the same shape. Note: quadrilaterals might not be convex.

It is impossible to completely tile the plane using only regular pentagons. However, can you identify at least three different types of pentagons (each with at least two different corresponding sides AND angles) that can be used to tile the plane in three distinct ways? Here essentially different means the tilings have different patterns.

Draw how to tile the whole plane with figures, made from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), where squares are used the same number of times in the design of the figure.

Show how to cover the plane with triangles of the following shape.

Draw how to tile the whole plane with figures, made from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), and \(4\times 4\), where squares are used the same amount of times in the design of the figure.

Show how to cover the plane with convex quadrilaterals.

Draw the plane tiling with:

• squares;

• rectangles \(1\times 3\);

• regular triangles;

• regular hexagons.

Draw the plane tiling using trapeziums of the following shape:

Here the sides \(AB\) and \(CD\) are parallel.