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?
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?
What is the best way to shunt these carriages so that each train can continue its journey?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
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?
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?
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?
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 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?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
Can you work out how to balance this equaliser? You can put more than one weight on a hook.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Can you cover the camel with these pieces?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
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?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
What happens when you try and fit the triomino pieces into these two grids?
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?
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?
Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.
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?
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
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 find all the different ways of lining up these Cuisenaire rods?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.
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....?
How many triangles can you make on the 3 by 3 pegboard?
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?
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.
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.
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.
How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
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?
Investigate the different ways you could split up these rooms so that you have double the number.
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
How many different triangles can you make on a circular pegboard that has nine pegs?