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...

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

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.

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

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?

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.

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

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?

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

Is there an efficient way to work out how many factors a large number has?

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

Find at least one way to put in some operation signs (+ - x ÷) to make these digits come to 100.

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. . . .

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?

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

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?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

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 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?

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

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

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

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?

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?

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

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

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

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

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

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. . . .

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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?

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

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

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

Can all unit fractions be written as the sum of two unit fractions?

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.

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

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?

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?