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 happens when you try and fit the triomino pieces into these two grids?

An activity making various patterns with 2 x 1 rectangular tiles.

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?

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?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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?

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?

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 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?

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?

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. . . .

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 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.

What is the best way to shunt these carriages so that each train can continue its journey?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

How many models can you find which obey these rules?

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.

These practical challenges are all about making a 'tray' and covering it with paper.

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.

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

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.

Investigate the different ways you could split up these rooms so that you have double the number.

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?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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.

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

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?

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

Can you find out in which order the children are standing in this line?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

Moira is late for school. What is the shortest route she can take from the school gates to the entrance?

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

If you had 36 cubes, what different cuboids could you make?

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?

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?

These two group activities use mathematical reasoning - one is numerical, one geometric.

Can you find all the different ways of lining up these Cuisenaire rods?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.