Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?

How many solutions can you find to this sum? Each of the different letters stands for a different number.

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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

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.

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?

A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?

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?

If a sum invested gains 10% each year how long before it has doubled its value?

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?

Explore the effect of reflecting in two parallel mirror lines.

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

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

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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

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?

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

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?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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...

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Use the differences to find the solution to this Sudoku.

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?

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

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 ?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

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?

Which of these games would you play to give yourself the best possible chance of winning a prize?

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

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?

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?