Explaining, convincing and proving

problem

What's a Group?

Age

16 to 18

Challenge level

2 out of 3

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.
problem

Golden Eggs

Age

16 to 18

Challenge level

2 out of 3

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

Fibonacci Fashion

Age

16 to 18

Challenge level

2 out of 3

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

Pythagorean Fibs

Age

16 to 18

Challenge level

2 out of 3

What have Fibonacci numbers got to do with Pythagorean triples?
problem

Integral Sandwich

Age

16 to 18

Challenge level

2 out of 3

Generalise this inequality involving integrals.
problem

Impossible Square?

Age

16 to 18

Challenge level

2 out of 3

Can you make a square from these triangles?
problem

Impossible Triangles?

Age

16 to 18

Challenge level

2 out of 3

Which of these triangular jigsaws are impossible to finish?
$Mind your \Ps and \Qs$

problem

Mind Your Ps and Qs

Age

16 to 18

Challenge level

2 out of 3

Sort these mathematical propositions into a series of 8 correct statements.
problem

Farey Neighbours

Age

16 to 18

Challenge level

2 out of 3

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?
problem

The Clue Is in the Question

Age

16 to 18

Challenge level

2 out of 3

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?