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.

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

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?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

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

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Investigate the different ways you could split up these rooms so that you have double the number.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do 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.

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.

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

What is the best way to shunt these carriages so that each train can continue its journey?

How many models can you find which obey these rules?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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.

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?

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

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

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

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

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

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

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....?

Can you find the chosen number from the grid using the clues?

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

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

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

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

An activity making various patterns with 2 x 1 rectangular tiles.