How many different ways can you find of fitting five hexagons together? How will you know you have found all the 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 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?
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?
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?
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?
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?
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?
What is the best way to shunt these carriages so that each train can continue its journey?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
How will you go about finding all the jigsaw pieces that have one peg and one hole?
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'.
Find your way through the grid starting at 2 and following these operations. What number do you end on?
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. . . .
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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?
What happens when you try and fit the triomino pieces into these two grids?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
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?
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?
Can you find ways of joining cubes together so that 28 faces are visible?
Have a go at this 3D extension to the Pebbles problem.
Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?
Can you work out what kind of rotation produced this pattern of pegs in our pegboard?
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?
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 relationship between these first two shapes? Which shape relates to the third one in the same way? Can you explain why?
A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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 is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Can you picture where this letter "F" will be on the grid if you flip it in these different ways?
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?
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?
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?
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Can you find a way of counting the spheres in these arrangements?
Can you visualise what shape this piece of paper will make when it is folded?