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?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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 many winning lines can you make in a three-dimensional version of noughts and crosses?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you find a rule which connects consecutive triangular numbers?
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. . . .
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Show that all pentagonal numbers are one third of a triangular number.
Can you find a rule which relates triangular numbers to square numbers?
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you use the diagram to prove the AM-GM inequality?
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. . . .
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
How good are you at finding the formula for a number pattern ?
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. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Surprising numerical patterns can be explained using algebra and diagrams...
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.
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
How to build your own magic squares.
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
Can you find the area of a parallelogram defined by two vectors?
A task which depends on members of the group noticing the needs of others and responding.
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
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. . . .
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?
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?
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
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.
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?
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. . . .
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.
If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?
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.
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.
The answer is $5x+8y$... What was the question?
Can you explain how this card trick works?
If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?
Can you make sense of these three proofs of Pythagoras' Theorem?
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
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 ?