In this article, the NRICH team describe the process of selecting solutions for publication on the site.

This article for primary teachers suggests ways in which to help children become better at working systematically.

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.

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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

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

How many different triangles can you make on a circular pegboard that has nine pegs?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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.

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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.

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

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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

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?

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.

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....?

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

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

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

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

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

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.

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

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

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 numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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

Can you find all the different ways of lining up these Cuisenaire rods?

In your bank, you have three types of coins. The number of spots shows how much they are worth. Can you choose coins to exchange with the groups given to make the same total?

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

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

Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

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?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

What is the smallest number of coins needed to make up 12 dollars and 83 cents?

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?

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

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

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

Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

How many different rhythms can you make by putting two drums on the wheel?