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?

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.

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?

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?

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

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

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.

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

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?

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?

How many models can you find which obey these rules?

In how many ways can you stack these rods, following the rules?

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?

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?

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?

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?

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?

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

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

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

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

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?

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

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

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?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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

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

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

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?

Place the numbers 1 to 10 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.

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

Place the numbers 1 to 6 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?

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

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

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.

Imagine that the puzzle pieces of a jigsaw are roughly a rectangular shape and all the same size. How many different puzzle pieces could there be?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

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?

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?