Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

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

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

If you move the tiles around, can you make squares with different coloured edges?

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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?

Which set of numbers that add to 10 have the largest product?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

There are lots of different methods to find out what the shapes are worth - how many can you find?

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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

Six balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

How well can you estimate 10 seconds? Investigate with our timing tool.

Play around with sets of five numbers and see what you can discover about different types of average...

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Where should you start, if you want to finish back where you started?

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

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?

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

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

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

7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

Which countries have the most naturally athletic populations?

Can you find a way to identify times tables after they have been shifted up?

There are nasty versions of this dice game but we'll start with the nice ones...

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".