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

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

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.

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.

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

A Sudoku with clues given as sums of entries.

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

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

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

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.

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

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

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

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

What could the half time scores have been in these Olympic hockey matches?

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

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?

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

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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?

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?

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.

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

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

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

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.

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

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?

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?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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

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

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

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.

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

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....?

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

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.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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

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

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 make on a circular pegboard that has nine pegs?

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

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?