Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

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?

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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 semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

Can you make sense of these three proofs of Pythagoras' Theorem?

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

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. . . .

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 task which depends on members of the group noticing the needs of others and responding.

How good are you at finding the formula for a number pattern ?

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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

Surprising numerical patterns can be explained using algebra and diagrams...

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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?

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

Can you find a rule which connects consecutive triangular numbers?

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

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

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

What is the total number of squares that can be made on a 5 by 5 geoboard?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Kyle and his teacher disagree about his test score - who is right?

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

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

Can you explain what is going on in these puzzling number tricks?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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