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.

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.

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.

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?

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.

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?

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.

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.

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.

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Arrange the digits 1, 1, 2, 2, 3 and 3 so that between the two 1's there is one digit, between the two 2's there are two digits, and between the two 3's there are three digits.

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.

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

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

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.

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

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

A Sudoku with clues given as sums of entries.

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?

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

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

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

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?

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

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

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.

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.

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

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

The Red Express Train usually has five red carriages. How many ways can you find to add two blue carriages?

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?

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?

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

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?

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

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?

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?

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

What happens when you try and fit the triomino pieces into these two grids?

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

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

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.