What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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.
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?
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?
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?
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?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
How will you go about finding all the jigsaw pieces that have one peg and one hole?
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?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
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?
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?
How many triangles can you make on the 3 by 3 pegboard?
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?
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.
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
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?
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 models can you find which obey these rules?
If you had 36 cubes, what different cuboids could you make?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
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?
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. . . .
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?
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?
This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
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?
Investigate the different ways you could split up these rooms so that you have double the number.
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.
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.
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?
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?
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
What could the half time scores have been in these Olympic hockey matches?
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?
Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?