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.

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

Explore the effect of reflecting in two parallel mirror lines.

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

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

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

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?

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.

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.

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

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

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

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

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

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

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?

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?

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?

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?

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

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

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

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

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?

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

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

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...

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

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

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

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

Can you find the area of a parallelogram defined by two vectors?

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

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. . . .

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

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

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

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?

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

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

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

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?

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

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

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?

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?