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

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

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

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

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

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?

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?

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

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?

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.

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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

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?

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

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?

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

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?

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. . . .

Explore the effect of reflecting in two parallel mirror lines.

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

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

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?

Explore the effect of combining enlargements.

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

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

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

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?

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

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

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

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

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. . . .

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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?

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?

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

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

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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?

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?

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?

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

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?