How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

This train line has two tracks which cross at different points. Can you find all the routes that end at Cheston?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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

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?

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?

My dice has inky marks on each face. Can you find the route it has taken? What does each face look like?

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

There are lots of different methods to find out what the shapes are worth - how many can you find?

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

The Zargoes use almost the same alphabet as English. What does this birthday message say?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

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?

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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?

Lorenzie was packing his bag for a school trip. He packed four shirts and three pairs of pants. "I will be able to have a different outfit each day", he said. How many days will Lorenzie be away?

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

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

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

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 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?

Find out what a "fault-free" rectangle is and try to make some of your own.

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

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?

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.

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

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?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

An investigation that gives you the opportunity to make and justify predictions.

If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?