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?

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

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

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?

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?

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

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

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.

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

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.

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

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

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. . . .

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.

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

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

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

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?

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

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

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

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.

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?

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

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?

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.

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

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

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

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?

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 ?

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

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

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

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

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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

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

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