Rational and irrational numbers

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 ?

Can you make a square from these triangles?

Which of these triangular jigsaws are impossible to finish?