Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

What is the smallest number of answers you need to reveal in order to work out the missing headers?

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

Can you find the hidden factors which multiply together to produce each quadratic expression?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Can you find the values at the vertices when you know the values on the edges?

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?

Play the game of Poison, Antidote, Water to start to understand the mathematics of associativity and groups.

A weekly challenge: these are shorter problems aimed at Post-16 students or enthusiastic younger students.

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.

Can you work out what simple structures have been dressed up in these advanced mathematical representations?

You had to work systematically to solve this problem although we received several different ways of doing so.

We received two very good solutions to this problem - one identified an important relationship, the other proved it.

We received both numerical and algebraic solutions to this problem. Lots of you made a generalisation and there were some clear proofs too.

John Mason describes the thinking behind this month's tasks.

The computer starts with all the lights off, but then clicks 3, 4 or 5 times at random, leaving some lights on. Can you switch them off again?

Can you write a programme to draw these seasonal shapes? Don't forget there's a limit on the number of symbols you can use!

We think this 3x3 version of the game is often harder than the 5x5 version. Do you agree? If so, why do you think that might be?

An introduction to the sort of algebra studied at university, focussing on groups.