Reasoning, convincing and proving

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}.

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Relate these algebraic expressions to geometrical diagrams.

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.