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Bored with their spiral-shaped yellow brick road, the Munchkins have decided to build a new, more angular, road, coloured red and blue and laid out using a cartesian coordinate system.
You have been asked to design some possible new roads, but must follow these very particular rules laid down by the Munchkins:
0. The road is to be built on a planar cartesian coordinate system.
1. Roads are built entirely from red equilateral triangle tiles and blue square tiles, all of side length one unit.
2. Tiles in a road must be joined exactly along edges with no overlap.
3. Triangular tiles must have an edge parallel to the $x$-axis.
4. In a finished road, all tiles except the start tile and end tile must be joined along an edge to exactly 2 other tiles.
4. A 'start tile' is a blue square joined on exactly one edge with a vertex at $(0, 0)$. Each road must contain a unique start tile.
5. An 'end tile' is a red triangle joined on exactly one edge. Each road must contain a unique end tile. The coordinates of the point on this triangle opposite this attached edge is called the destination of the path.
Can you detemine which of these roads could satisfy the Munchkins' rules given a coordinate system of your choice?
How many roads which would not satisfy EXACTLY ONE of the Munchkins' rules can you make using 2, 3 or 4 tiles?
A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?