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

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

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

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

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

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

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 numbers from a sequence. What numbers can we make now?

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?

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

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

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

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

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?

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

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

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?

Six balls are shaken. You win if at least one red ball ends in a corner. 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 is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

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

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

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

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 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?

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?

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

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

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?

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

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

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?

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

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?

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

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

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

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

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

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

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

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

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

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

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.