Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
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?
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?
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?
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?
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.
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 could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?
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?
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?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
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....?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Can you find all the different ways of lining up these Cuisenaire rods?
Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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 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?
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?
How many trapeziums, of various sizes, are hidden in this picture?
The Zargoes use almost the same alphabet as English. What does this birthday message say?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?
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.
When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.
Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.
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?
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.
Investigate the different ways you could split up these rooms so that you have double the number.
Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.
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?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
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?
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?
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.
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?