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.

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.

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

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.

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

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.

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

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

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

A challenging activity focusing on finding all possible ways of stacking rods.

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?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

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

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

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

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?

What two-digit numbers can you make with these two dice? What can't you make?

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

This challenge extends the Plants investigation so now four or more children are involved.

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

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

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

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

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

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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?

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

Find out what a "fault-free" rectangle is and try to make some 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?

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

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?

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

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

Can you fill in the empty boxes in the grid with the right shape and colour?

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?

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

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

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

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?

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

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?

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?

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