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.

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

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?

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.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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.

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?

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

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

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

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

A Sudoku with clues given as sums of entries.

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

How many solutions can you find to this sum? Each of the different letters stands for a different number.

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

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

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.

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?

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

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?

A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . .

Find out about Magic Squares in this article written for students. Why are they magic?!

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

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 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?

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

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

Can you find all the different triangles on these peg boards, and find their angles?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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?

Try out the lottery that is played in a far-away land. What is the chance of winning?

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

In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

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.

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?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

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

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

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?