Explaining, convincing and proving

problem
Is it magic or is it maths?

Age

11 to 14

Challenge level

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying them. Can you explain how they work?
problem
Königsberg

Age

11 to 14

Challenge level

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
problem
Clocked

Age

11 to 14

Challenge level

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
problem
Circle box

Age

14 to 16

Challenge level

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
problem
Generally geometric

Age

16 to 18

Challenge level

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
problem
Cube net

Age

16 to 18

Challenge level

How many tours visit each vertex of a cube once and only once? How many return to the starting point?
problem
Can it be?

Age

16 to 18

Challenge level

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
problem
And so on - and on - and on

Age

16 to 18

Challenge level

Can you find the value of this function involving algebraic fractions for x=2000?
problem
A leg to stand on

Age

11 to 14

Challenge level

Can you work out the number of chairs at a cafe from the number of legs?
problem
Discrete trends

Age

16 to 18

Challenge level

Find the maximum value of n to the power 1/n and prove that it is a maximum.