### 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 2
Students 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?

Possible Support

Starting from a blue square, ask students to build up valid roads
using the rules. As they build roads, can they see where problems
might occur?