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.
In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Find the sum of all three-digit numbers each of whose digits is
A collection of games on the NIM theme
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
Here are two kinds of spirals for you to explore. What do you notice?
Find out what a "fault-free" rectangle is and try to make some of
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'.
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
An investigation that gives you the opportunity to make and justify
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
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
Can all unit fractions be written as the sum of two unit fractions?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Explore the effect of reflecting in two intersecting mirror lines.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
Explore the effect of combining enlargements.
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
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?
Can you tangle yourself up and reach any fraction?
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?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
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?
Can you find the values at the vertices when you know the values on
It would be nice to have a strategy for disentangling any tangled
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
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
What happens when you round these three-digit numbers to the nearest 100?
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
This task follows on from Build it Up and takes the ideas into three dimensions!