Road maker

We received two particularly full and clear solutions to this problem. Both submitters thought carefully about the precise mathematical meaning of the rules. The first solution was from Patrick, from Woodbridge School and the second from Phil at Garforth Community College. 

Image

1. The road is to be built on a planar cartesian coordinate system.
   
    
2. Roads are built entirely from red equilateral triangle tiles and blue square tiles, all of side length one unit.
   
    
3. Tiles in a road must be joined exactly along edges with no overlap.
   
    
4. Triangular tiles must have an edge parallel to the $x$-axis.
   
    
5. 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.
   
    
6. 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.
   
    
7. In a finished road, all tiles except the start tile and end tile must be joined along an edge to exactly 2 other tiles.

First solution from Patrick:

These roads obey the all rules:

A, B, E, H

These roads disobey the rules:

C disobeys rules 5, 6 (it has two end tiles and no start tile)

D disobeys rules 5, 6 (no start or end tiles)

F disobeys rule 4 (a triangular tile does not have any of it's edges parallel to the $x$ axis)

G disobeys rule 5, 7 (no start tile)

I disobeys rule 6 (no end tile)

J disobeys rules 5, 6 (no start tile and two end tiles)

K disobeys rule 5 (no start tile - start tile has to have edges on the $x$ and $y$ axes)

L disobeys rules 5, 6 

M disobeys rule 4 (a triangular tile does not have any of it's edges parallel to the $x$ axis)

Phil from Garforth Community College gave a clear explanation as to which roads satisfied or broke each rule. He thought like a true mathematician, questioning all rules carefully. We were particularly pleased that he noted that 'touching corners' might count as an overlap; this would require proper clarification.

The following roads fit all seven rules: A, B, E and H .

These roads break the rules as follows:

Rule 3: All the tiles seem to adhere to this law, although it could be argued that touching corners such as the squares in H count as an overlap .

Rule 4: F and M disobey this rule.

Rule 5: It could be argued that L, D and C break this rule, because there appears to be no distinct start tile. It is ambiguous whether a distinct start is needed, which only connects to one other tile, or simply a tile which could be designated as the start .

Rule 6: I breaks this rule, by having no red triangles, although D does not appear to have a distinct destination, leading to a similar confusion as in rule one .

Rule 7: In G, there is one square attached to three tiles .

Phil continued to analyse some more of the structure of the problem:

Theoretically, there are in fact an infinite number of paths which break only one of the rules. This comes from a loophole found in rule number three:

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

The following road would comply to the munchkins' standards. Yes; this is the shortest road

Image

Image