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?

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

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

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?

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?

Explore the effect of reflecting in two parallel mirror lines.

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

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?

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

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?

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?

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

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

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?

Explore the effect of combining enlargements.

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

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

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

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

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

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?

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?

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.

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.

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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

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?

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

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

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. . . .

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?

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

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

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?

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?

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

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?

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

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.

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?