Reasoning, convincing and proving

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage stamps? Prove that all other values can be made up.

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.

Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?

The 30 students in a class have 25 different birthdays between them. What is the largest number that can share any birthday?

Which of these roads will satisfy a Munchkin builder?

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

How do these modelling assumption affect the solutions?

Four jewellers share their stock. Can you work out the relative values of their gems?

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.