Rational and irrational numbers

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

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

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

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.

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

Can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?

What fractions can you find between the square roots of 65 and 67?

If the yellow equilateral triangle is taken as the unit for area, what size is the hole ?