These pictures show some different activities that you may get up to during a day. What order would you do them in?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

These pictures show squares split into halves. Can you find other ways?

In this game you throw two dice and find their total, then move the appropriate counter to the right. Which counter reaches the purple box first? Is this what you would expect?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

How many different sets of numbers with at least four members can you find in the numbers in this box?

There are three buckets each of which holds a maximum of 5 litres. Use the clues to work out how much liquid there is in each bucket.

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Which of these games would you play to give yourself the best possible chance of winning a prize?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Can you make sense of these three proofs of Pythagoras' Theorem?

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

Why MUST these statistical statements probably be at least a little bit wrong?

Was it possible that this dangerous driving penalty was issued in error?

Have a go at being mathematically negative, by negating these statements.

Did you break the spells? We had some very clear explanations of how to complete these sequences.

This problem was very well answered by many of you using a systematic approach. You knew that you hadn't missed out any possibilities because of the way you ordered the solutions. Well done!

Harry sent us a very clear algebraic solution to this problem, and we also received lots of ideas on how to solve it numerically.

Is the process fair? This question often gives rise to disagreements and discussion. Tom gives a clear logical explanation and uses a tree diagram and a spreadsheet.

This fascinating article delves into the world of talk in the classroom and explains how an understanding of talking can really improve the learning of mathematics.

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

An article demonstrating mathematically how various physical modelling assumptions affect the solution to the seemingly simple problem of the projectile.