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# Road Maker

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Age 14 to 18

Challenge Level

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?**