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?

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

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.

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?

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?

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

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?

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

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

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

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

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?

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

How many different shapes can you make by putting four right- angled isosceles triangles together?

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

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

Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?

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

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

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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.

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?

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

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

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

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?

Your challenge is to find the longest way through the network following this rule. You can start and finish anywhere, and with any shape, as long as you follow the correct order.

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?

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

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

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

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?

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?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Try this matching game which will help you recognise different ways of saying the same time interval.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

What happens when you round these three-digit numbers to the nearest 100?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

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?

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?

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?

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?

The brown frog and green frog want to swap places without getting wet. They can hop onto a lily pad next to them, or hop over each other. How could they do it?

Take three differently coloured blocks - maybe red, yellow and blue. Make a tower using one of each colour. How many different towers can you make?

This challenge is about finding the difference between numbers which have the same tens digit.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

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?

This task follows on from Build it Up and takes the ideas into three dimensions!