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?
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.
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?
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.
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?
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?
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
Can you make sense of these three proofs of Pythagoras' Theorem?
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.
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
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. . . .
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?
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. . . .
A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?
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. . . .
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?
How good are you at finding the formula for a number pattern ?
How to build your own magic squares.
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?
Surprising numerical patterns can be explained using algebra and diagrams...
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. . . .
Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?
Can you explain how this card trick works?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?
A task which depends on members of the group noticing the needs of others and responding.
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
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.
Can you explain why a sequence of operations always gives you perfect squares?
Can you use the diagram to prove the AM-GM inequality?
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Can you explain what is going on in these puzzling number tricks?
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
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?
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
What is the total number of squares that can be made on a 5 by 5 geoboard?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Can you find the lap times of the two cyclists travelling at constant speeds?
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 ï¿½ 1 [1/3]. What other numbers have the sum equal to the product and can this be. . . .
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. . . .