Can you cover the camel with these pieces?
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?
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.
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 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?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
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 smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
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?
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?
A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?
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?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
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 ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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?
What happens when you try and fit the triomino pieces into these two grids?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
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?
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?
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?
Can you describe a piece of paper clearly enough for your partner to know which piece it is?
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?
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
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?
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?
Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?
Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?
A variant on the game Alquerque
Move just three of the circles so that the triangle faces in the opposite direction.
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 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 many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
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?
How many balls of modelling clay and how many straws does it take to make these skeleton shapes?
Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?
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?
Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.
Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?
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?
We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?