Problems

Show how to cover the plane with convex quadrilaterals.

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.

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.

For any triangle, prove you can tile the plane with that triangle.

Prove that one cannot tile the whole plane with regular pentagons.

Draw the plane tiling using convex hexagons with parallel and equal opposite sides.

Draw how to tile the whole plane with figures, consisting of squares \(1\times 1\), \(2\times 2\), \(3\times 3\), \(4\times 4\), \(5\times 5\), and \(6\times 6\), where each square appears an equal number of times in the design of the figure. Can you think of two essentially different ways to do this?

Find a non-regular octagon which you can use to tile the whole plane and show how to do that.

A round-robin tournament is one where each team plays every other team exactly once. Five teams take part in such a tournament getting: \(3\) points for a win, \(1\) point for a draw and \(0\) points for a loss. At the end of the tournament the teams are ranked from first to last according to the number of points.
Is it possible that at the end of the tournament, each team has a different number of points, and each team except for the team ranked last has exactly two more points than the next-ranked team?