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

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

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 difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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

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

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

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

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

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?

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

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

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

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

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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

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

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

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

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?

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?

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?

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

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

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

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

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?

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

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?

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

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

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?

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

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

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?

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

Which countries have the most naturally athletic populations?

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.

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

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

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

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?

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

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

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

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