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

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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

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

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.

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?

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. . . .

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?

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

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 ?

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

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

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?

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.

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

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 ?

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

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

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

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

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

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

Which of these games would you play to give yourself the best possible chance of winning a prize?

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 ?

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

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?

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. . . .

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of. . . .

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

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

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 the area of a parallelogram defined by two vectors?

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?

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.

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?

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?

Use the differences to find the solution to this Sudoku.

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

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

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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.

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?