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?

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

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

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

Which countries have the most naturally athletic populations?

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?

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

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?

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

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

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Use your skill and judgement to match the sets of random data.

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...

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

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

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

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?

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

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

What's the largest volume of box you can make from a square of paper?

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

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

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?

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

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...

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?

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?

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

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

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?

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

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

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

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

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

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

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

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

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?

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.

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

Can you work out which spinners were used to generate the frequency charts?

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

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?

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

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