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?
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.
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?
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 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?
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?
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. . . .
Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?
Use the clues to colour each square.
This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?
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?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
An activity making various patterns with 2 x 1 rectangular tiles.
What happens when you try and fit the triomino pieces into these two grids?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
These practical challenges are all about making a 'tray' and covering it with paper.
Can you cover the camel with these pieces?
Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
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 many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
Can you work out how to balance this equaliser? You can put more than one weight on a hook.
Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Can you find all the different ways of lining up these Cuisenaire rods?
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?
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.
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?
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?
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?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
Investigate the different ways you could split up these rooms so that you have double the number.
How many triangles can you make on the 3 by 3 pegboard?
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.
The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
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?
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
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 can you put five cereal packets together to make different shapes if you must put them face-to-face?
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?