Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
What is the best way to shunt these carriages so that each train can continue its journey?
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?
We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
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?
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?
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?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?
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 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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
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 make a 3x3 cube with these shapes made from small cubes?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
Can you cover the camel with these pieces?
What happens when you try and fit the triomino pieces into these two grids?
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?
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.
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?
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?
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?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
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?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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?
Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?
Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?
Move just three of the circles so that the triangle faces in the opposite direction.
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 quadrilaterals can be made by joining the dots on the 8-point circle?
A variant on the game Alquerque
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?
A game for two players. You'll need some counters.
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.
How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 cube that is green all over AND a 2 x 2 cube that is yellow all over?
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 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?
Exchange the positions of the two sets of counters in the least possible number of moves
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 game for two players on a large squared space.
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?
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?