A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

Which of these triangular jigsaws are impossible to finish?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Can you describe this route to infinity? Where will the arrows take you next?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

Look for the common features in these graphs. Which graphs belong together?

In this short problem, can you deduce the likely location of the odd ones out in six sets of random numbers?

Can you make matrices which will fix one lucky vector and crush another to zero?

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

When does a pattern start to exhibit structure? Can you crack the code used by the computer?