Reasoning, justifying, convincing and proof - advanced

Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.

Infographics are a powerful way of communicating statistical information. Can you come up with your own?

What do you get when you raise a quadratic to the power of a quadratic?

What happens when you find the powers of this matrix?

Can you prove these triangle theorems both ways?

Can you find out what numbers divide these expressions? Can you prove that they are always divisors?

Can you work out which spinners were used to generate the frequency charts?

Can you rearrange the cards to make a series of correct mathematical statements?

Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

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.

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

Which numbers cannot be written as the sum of two or more consecutive numbers?

This problem challenges you to find cubic equations which satisfy different conditions.

$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?

10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

Can you prove that triangles are right-angled when $a^2+b^2=c^2$?

What does this number mean? Which order of 1, 2, 3 and 4 makes the highest value? Which makes the lowest?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

Can you make sense of the three methods to work out what fraction of the total area is shaded?

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

Have a go at being mathematically negative, by negating these statements.

Here you have an opportunity to explore the proofs of the laws of logarithms.

Can you match the charts of these functions to the charts of their integrals?

Was it possible that this dangerous driving penalty was issued in error?

Which of the statements is true?

This is a beautiful result involving a parabola and parallels.

Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Which of the cards provides the counter example?

Which statement correctly describes the real roots of the equation?

Change the squares in this diagram and spot the property that stays the same for the triangles. Explain...

What can you say about the common difference of an AP where every term is prime?

If the statement is not true, what can we say will be true?

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

Can you prove that a quadrilateral drawn inside a tetrahedron is a parallelogram?

Can you find a way to prove the trig identities using a diagram?

In this short challenge, can you use angle properties in a circle to figure out some trig identities?

Can you prove our inequality holds for all values of x and y between 0 and 1?

What can you deduce about the gradients of curves linking (0,0), (8,8) and (4,6)?

There are many different methods to solve this geometrical problem - how many can you find?