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

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

A weekly challenge concerning powers and quadratic equations.

How good are you at finding the formula for a number pattern ?

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

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

Make a conjecture about the curved track taken by the complex roots of a quadratic equation and use complex conjugates to prove your conjecture.

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

Draw any triangle PQR. Find points A, B and C, one on each side of the triangle, such that the area of triangle ABC is a given fraction of the area of triangle PQR.

Exploit the symmetry and turn this quartic into a quadratic.

Can you find the maximum value of the curve defined by this expression?

Can you find a quadratic equation which passes close to these points?

Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

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

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

A bag contains red and blue balls. You are told the probabilities of drawing certain combinations of balls. Find how many red and how many blue balls there are in the bag.

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

Sketch the graphs for this implicitly defined family of functions.

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to. . . .

What is the same and what is different about these circle questions? What connections can you make?

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.

How many eggs should a bird lay to maximise the number of chicks that will hatch? An introduction to optimisation.

Find the equation from which to calculate the resistance of an infinite network of resistances.