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?

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

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

This challenge is about finding the difference between numbers which have the same tens digit.

What two-digit numbers can you make with these two dice? What can't you make?

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

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

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?

Find out about Magic Squares in this article written for students. Why are they magic?!

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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

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

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

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.

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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

In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

How many possible necklaces can you find? And how do you know you've found them all?

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?

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?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

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

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

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?

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?

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?

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?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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.

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?

Can you fill in the empty boxes in the grid with the right shape and colour?

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

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

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.

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?

Find all the numbers that can be made by adding the dots on two dice.

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?

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

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

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?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?