Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

This task follows on from Build it Up and takes the ideas into three dimensions!

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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.

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

What's the largest volume of box you can make from a square of paper?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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

An investigation that gives you the opportunity to make and justify predictions.

Here are two kinds of spirals for you to explore. What do you notice?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Can you find the values at the vertices when you know the values on the edges?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

This challenge asks you to imagine a snake coiling on itself.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

Surprise your friends with this magic square trick.

Watch this animation. What do you see? Can you explain why this happens?

Are these statements always true, sometimes true or never true?

Are these statements always true, sometimes true or never true?

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

These tasks give learners chance to generalise, which involves identifying an underlying structure.