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

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?

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?

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

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?

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

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

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

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

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

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

Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?

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

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

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

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

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

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

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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.

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

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?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

How many trapeziums, of various sizes, are hidden in this picture?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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.

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

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?

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?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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

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.

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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?

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?

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

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.

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

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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

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?