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?

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 how to balance this equaliser? You can put more than one weight on a hook.

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

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?

How many trains can you make which are the same length as Matt's, using rods that are identical?

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?

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....?

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?

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

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

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?

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

Design an arrangement of display boards in the school hall which fits the requirements of different people.

How many different rhythms can you make by putting two drums on the wheel?

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?

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.

In this matching game, you have to decide how long different events take.

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

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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 triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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

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

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

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

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?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

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?

These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.

How many different triangles can you make on a circular pegboard that has nine pegs?

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?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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.

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?

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

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?

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.

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?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have 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.

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

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.