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?

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

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 ?

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

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?

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

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

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?

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

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

Explore the effect of combining enlargements.

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

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

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?

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

Explore the effect of reflecting in two parallel mirror lines.

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?

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

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?

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

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

How many winning lines can you make in a three-dimensional version of noughts and crosses?

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

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

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

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.

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

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 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.

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. . . .

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

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

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

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

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.

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?

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

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?

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

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

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

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

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?

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

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

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