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?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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?
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?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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. . . .
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
How many different rhythms can you make by putting two drums on the wheel?
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 how many ways can you stack these rods, following the rules?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
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?
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?
A challenging activity focusing on finding all possible ways of stacking rods.
What happens when you try and fit the triomino pieces into these two grids?
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.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
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?
Can you cover the camel with these pieces?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
How many different triangles can you make on a circular pegboard that has nine pegs?
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
Use the clues to colour each square.
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.
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?
Investigate the different ways you could split up these rooms so that you have double the number.
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?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
Can you work out how to balance this equaliser? You can put more than one weight on a hook.
You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?
Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.
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?
Can you find all the different ways of lining up these Cuisenaire rods?
A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?
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 6 in the circles so that each number is the difference between the two numbers just below it.
Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?