A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Use the interactivity to find out how many quarter turns the man must rotate through to look like each of the pictures.

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

This ladybird is taking a walk round a triangle. Can you see how much he has turned when he gets back to where he started?

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

How many possible necklaces can you find? And how do you know you've found them all?

This problem is intended to get children to look really hard at something they will see many times in the next few months.