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

Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?

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.

Can you work out the probability of winning the Mathsland National Lottery? Try our simulator to test out your ideas.

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

Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?

If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?

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

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

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

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

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

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?

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

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?

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

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

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?

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

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

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

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

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".

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

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

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 have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

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

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

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

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

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?

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

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

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?

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

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?

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

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

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

Which countries have the most naturally athletic populations?

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

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

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

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

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?

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

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?