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.

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.

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?

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.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

Find the sum of all three-digit numbers each of whose digits is odd.

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

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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

Try out this number trick. What happens with different starting numbers? What do you notice?

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.

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.

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?

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?

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?

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

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.

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

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?

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

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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

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

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

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?

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

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?

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

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

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?

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?”

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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

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

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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

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?

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.

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Strike it Out game for an adult and child. Can you stop your partner from being able to go?