What is the best way to shunt these carriages so that each train can continue its journey?
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
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?
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?
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.
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
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?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
Can you find all the different ways of lining up these Cuisenaire rods?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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....?
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?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Can you find all the different triangles on these peg boards, and find their angles?
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?
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 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. . . .
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.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
How many triangles can you make on the 3 by 3 pegboard?
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?
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.
These practical challenges are all about making a 'tray' and covering it with paper.
Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
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?
Using the statements, can you work out how many of each type of rabbit there are in these pens?
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?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
In how many ways can you stack these rods, following the rules?
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
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?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?