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A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS? ### Be Reasonable

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression. ### Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### 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?