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

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

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.

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

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

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

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

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.

A weekly challenge concerning powers and quadratic equations.

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

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

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?

Exploit the symmetry and turn this quartic into a quadratic.

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. . . .

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.

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

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

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

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

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. . . .

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?

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.

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.

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?

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.

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.

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

Sketch the graphs for this implicitly defined family of functions.

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

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

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.

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

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 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.

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.