Reasoning, convincing and proving

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

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

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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?