### Be Reasonable

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

### Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

### Rational Roots

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

### Why do this problem?

Provide training in conjecture, mathematical analysis and proof. This difficult problem requires students to realise that it is possible to use counting (a discrete process) to somehow categorise different paths. This gives the power to make general statements.

### Possible Approach?

First students need to calculate the end points of a few simple roads. Although there are rational and irrational endpoints, group discussion should lead to the conclusion that root 3 should be involved in the irrational endpoints in some way.

### Key Questions?

If you have a valid road, how can its endpoint change with the addition of a single tile? Two tiles?

Can you make any 'families' of roads which are similar, yet of different lengths? Can you create expressions for the lengths of these families of roads?