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?

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

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

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?

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

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?

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

Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.

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?

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

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?

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

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.

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

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

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?

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?

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?

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?

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

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

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 ?

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

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?

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

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

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

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 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?

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?

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?

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

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

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

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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?

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. . . .

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?

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

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

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

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?

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 rectangles where the value of the area is the same as the value of the perimeter?

Explore the effect of reflecting in two parallel mirror lines.

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?

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

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