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

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

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

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

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. . . .

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?

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

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

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.

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

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

A Sudoku with clues given as sums of entries.

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?

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?

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

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.

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

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

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

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

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?

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?

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

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....?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

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?

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

How many trains can you make which are the same length as Matt's, using rods that are identical?

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

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

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

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

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?

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

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

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.

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

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

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

If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced?

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

In this matching game, you have to decide how long different events take.

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.