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. . . .
In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
Kyle and his teacher disagree about his test score - who is right?
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?
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?
Show that all pentagonal numbers are one third of a triangular number.
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 ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
Can you find a rule which connects consecutive triangular numbers?
Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
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.
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?
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?
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.
My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .
How good are you at finding the formula for a number pattern ?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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. . . .
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Can you make sense of these three proofs of Pythagoras' Theorem?
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?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
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?
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.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
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?
The Number Jumbler can always work out your chosen symbol. Can you work out how?
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. . . .
Can you explain what is going on in these puzzling number tricks?
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. . . .
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 use the diagram to prove the AM-GM inequality?
What is special about the difference between squares of numbers adjacent to multiples of three?
Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?