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.
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?
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.
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?
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?
How many models can you find which obey these rules?
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?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
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?
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?
In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?
These practical challenges are all about making a 'tray' and covering it with paper.
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
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?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.
Using the statements, can you work out how many of each type of rabbit there are in these pens?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
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?
What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?
There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.
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?
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?
Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.
In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Investigate the different ways you could split up these rooms so that you have double the number.
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.
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.
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?
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?
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.