Rational and irrational numbers

Find the sum of this series of surds.

Can you express every recurring decimal as a fraction?

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

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

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

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?

Can you make a square from these triangles?

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

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

Which of these triangular jigsaws are impossible to finish?