Problems

Terry and Janet are playing a game with stones. There are two piles of stones, one has \(m\) stones and the other has \(n\) stones initially. In their turn, a player takes from one pile a positive number of stones that is a multiple of the number of stones in the other pile at that moment. The player who cleans up one of the piles wins. Terry starts - who will win?

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

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?