Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?
Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
How will you go about finding all the jigsaw pieces that have one peg and one hole?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?
10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
What is the best way to shunt these carriages so that each train can continue its journey?
Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?
A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
Have a go at this 3D extension to the Pebbles problem.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?
What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?
When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.
Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?
Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?
In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?
A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?
Can you find ways of joining cubes together so that 28 faces are visible?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.
This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
How many different triangles can you make on a circular pegboard that has nine pegs?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?
Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?
Can you discover whether this is a fair game?
What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?
Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
Watch this animation. What do you see? Can you explain why this happens?
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?