# Road maker

## Problem

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?**

## Getting Started

## Student Solutions

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.

**For a reminder, the roads are as in the picture above and the rules are as follows:**

- 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.

**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

None of the following, however, would:

There is a start, a destination, the triangles point due north and both tiles are only attached to one other. Those are only four of the possible paths which break rule three, but the triangle can be shifted by smaller and smaller amounts each time, leading to a possible infinite number of combinations in the paths. This probing of the rules to see how they can be failed shows real mathematical insight -- well done!

## Teachers' Resources

### Why use this problem?

This problem trains students to work with formal systems of axioms, such as might be encountered in situations in discrete mathematics (such as group theory). It will help to clarify students' understanding of exact mathematical meaning, as opposed to normal language (which is inexact). This will help students to argue a mathematical point.### Possible Approach

Encourage students first to decide individually on which roads they believe are allowed. Then vote as a group on each road. There is likely to be some initial disagreement: students should be encouraged to argue their points, using the rules to back up their arguments.When a student presents an argument for or against a road, ask the rest of the class to decide whether they are using the rules precisely. For example, are they using only the stated properties of the 'start' tile, or are they also using extra meanings of 'start' implied by English language.### Key questions

Do you think that these rules are consistent?Are these rules precise enough in meaning?

In what ways does this mathematical description of a 'road' differ from your everyday conception of a 'road'? In what ways are they the same?

Would you want to clarify any rules or add any other rules? Why?

### Possible Extensions

Can you find the possible points at which roads can end starting with a square cornered at the origin? More details of this question are provided in the follow up question Road Maker 2Students interested in the ideas surrounding formal rules and axioms might like to read the article How Many Geometries Are There? or the article What is a group?