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

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?

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

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 ?

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

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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

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

If a sum invested gains 10% each year how long before it has doubled its value?

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

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.

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. . . .

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

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?

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?

Substitute -1, -2 or -3, into an algebraic expression and you'll get three results. Is it possible to tell in advance which of those three will be the largest ?

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

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?

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

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

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

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

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?

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

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.

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?

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

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

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 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?

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

Explore the effect of combining enlargements.

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.

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?

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

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

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?

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?

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

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

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

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

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