In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

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?

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?

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?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

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

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?

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?

My coat has three buttons. How many ways can you find to do up all the buttons?

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?

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.

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

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.

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

This challenge extends the Plants investigation so now four or more children are involved.

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

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?

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.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

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

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

How many models can you find which obey these rules?

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?

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

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

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as 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?

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

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?

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

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

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?

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

Can you fill in the empty boxes in the grid with the right shape and colour?

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

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

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?

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?

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?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Find all the numbers that can be made by adding the dots on two dice.

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

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

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