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

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?

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?

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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

Can you find the chosen number from the grid using the clues?

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

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

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

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?

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

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

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

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

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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?

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?

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

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

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

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

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

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

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

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

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?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

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

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

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

If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?

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

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

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

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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

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

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?

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

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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?