A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

Explore the relationships between different paper sizes.

Draw a square and an arc of a circle and construct the Golden rectangle. Find the value of the Golden Ratio.

Explain how to construct a regular pentagon accurately using a straight edge and compass.

What have Fibonacci numbers got to do with Pythagorean triples?

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?