Design an arrangement of display boards in the school hall which fits the requirements of different people.
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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?
What is the best way to shunt these carriages so that each train can continue its journey?
The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
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?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
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?
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?
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. . . .
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?
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?
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?
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
How many models can you find which obey these rules?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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 many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
How many ways can you find of tiling the square patio, using square tiles of different sizes?
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?
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?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?
Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?
Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?
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?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
An activity making various patterns with 2 x 1 rectangular tiles.
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
Can you find all the different ways of lining up these Cuisenaire rods?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Two sudokus in one. Challenge yourself to make the necessary connections.
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.
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?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
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?
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?