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?

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?

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

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

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?

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

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 ...?

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?

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.

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

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 if you have an ordered approach.

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

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?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

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?

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

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?

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 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?

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?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

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

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

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 task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

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'.

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

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

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

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

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?

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.

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?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

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

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

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

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?

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

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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

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?