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

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?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Can you describe this route to infinity? Where will the arrows take you next?

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

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

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

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

Can you maximise the area available to a grazing goat?

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

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

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?

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?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

How many different symmetrical shapes can you make by shading triangles or squares?

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

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

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".

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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?

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

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

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

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

The clues for this Sudoku are the product of the numbers in adjacent squares.