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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:

- The road is to be built on a planar cartesian coordinate system.

- Roads are built entirely from red equilateral triangle tiles and blue square tiles, all of side length one unit.

- Tiles in a road must be joined exactly along edges with no overlap.

- Triangular tiles must have an edge parallel to the $x$-axis.

- A 'start tile' is a blue square with a vertex at $(0, 0)$ and with an edge which lies on the $x$ and and edge which lies on the $y$ axes. Each road must contain a unique start tile, and the start tile is joined on exactly one edge.

- An 'end tile' is a red triangle joined on exactly one edge. Each road must contain a unique end tile. The point on this triangle opposite this attached edge is called the
**destination**of the path.

- In a finished road, all tiles except the start tile and end tile must be joined along an edge to exactly 2 other tiles.

**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?**