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?

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?

Which countries have the most naturally athletic populations?

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

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?

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

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

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

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.

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

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

You'll need to know your number properties to win a game of Statement Snap...

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

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

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

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?

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

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

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?

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

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

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

Can you find any two-digit numbers that satisfy all of these statements?

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

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

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

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

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

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

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

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

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?

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

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

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

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?

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?

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

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

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

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

In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.

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

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?

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

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".