What happens when you try and fit the triomino pieces into these two grids?
How will you go about finding all the jigsaw pieces that have one peg and one hole?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
Can you cover the camel with these pieces?
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?
Move just three of the circles so that the triangle faces in the opposite direction.
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?
You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?
A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
Make one big triangle so the numbers that touch on the small triangles add to 10.
Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?
A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
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?
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
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?
What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?
Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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. . . .
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?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
Find your way through the grid starting at 2 and following these operations. What number do you end on?
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'.
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?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
What is the greatest number of squares you can make by overlapping three squares?
Have a go at this 3D extension to the Pebbles problem.
What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.
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?
A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.
Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?
What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?