If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

My coat has three buttons. How many ways can you find to do up all the buttons?

Dotty Six is a simple dice game that you can adapt in many ways.

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

When dice land edge-up, we usually roll again. But what if we didn't...?

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

Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

Photocopiers can reduce from A3 to A4 without distorting the image. Explore the relationships between different paper sizes that make this possible.

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

Can you devise a fair scoring system when dice land edge-up or corner-up?

How did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

You used excellent reasoning based on your own school circumstances to answer this problem.

Many of you explained clearly why the second version of the lottery was harder than the first and we received some interesting lottery games of your own creation.

Do you want to understand what your chances are of winning the lottery? Then take a look at these solutions.

Analysis of this problem is fascinating because it draws together a heady mix of theoretical and numerical probability along with estimates and approximations.

Jenny Piggott reflects on the event held to mark her retirement from the directorship of NRICH, but also on problem solving itself.

These models have appeared around the Centre for Mathematical Sciences. Perhaps you would like to try to make some similar models of your own.

Unmultiply is a game of quick estimation. You need to find two numbers that multiply together to something close to the given target - fast! 10 levels with a high scores table.

This article discusses what happens, and why, if you generate chains of sequences getting the next sequence from the differences between the adjacent terms in the sequence before it, eg (7, 2, 8, 3) maps to (5, 6, 5, 4).