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?

Think of two whole numbers under 10. Take one of them and add 1. Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your second number. Add 2. Double again. Subtract 8. Halve this number. . . .

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

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?

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?

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?

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

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

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

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

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

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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

There are lots of different methods to find out what the shapes are worth - how many can you find?

Use the differences to find the solution to this Sudoku.

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?

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

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. . . .

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

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?

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

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

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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?

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

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?

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

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 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?

What is the same and what is different about these circle questions? What connections can you make?

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?

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

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

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

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

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

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. . . .

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. . . .

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

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

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

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

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?

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