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?
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?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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?
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?
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?
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?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
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.
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?
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?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
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?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
How many triangles can you make on the 3 by 3 pegboard?
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?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
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?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
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?
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....?
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?
Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
How many different triangles can you make on a circular pegboard that has nine pegs?
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.
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.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Can you find all the different ways of lining up these Cuisenaire rods?
Investigate the different ways you could split up these rooms so that you have double the number.
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?