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.
Can you discover whether this is a fair game?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
A huge wheel is rolling past your window. What do you see?
Some puzzles requiring no knowledge of knot theory, just a careful
inspection of the patterns. A glimpse of the classification of
knots and a little about prime knots, crossing numbers and. . . .
Do you know how to find the area of a triangle? You can count the
squares. What happens if we turn the triangle on end? Press the
button and see. Try counting the number of units in the triangle
now. . . .
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?
Can you find all the 4-ball shuffles?
This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
Is it true that any convex hexagon will tessellate if it has a pair
of opposite sides that are equal, and three adjacent angles that
add up to 360 degrees?
You have been given nine weights, one of which is slightly heavier
than the rest. Can you work out which weight is heavier in just two
weighings of the balance?
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.
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
This is the second article on right-angled triangles whose edge lengths are whole numbers.
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
Investigate circuits and record your findings in this simple introduction to truth tables and logic.
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .
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?
A introduction to how patterns can be deceiving, and what is and is not a proof.
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Use your logical reasoning to work out how many cows and how many
sheep there are in each field.
This article stems from research on the teaching of proof and
offers guidance on how to move learners from focussing on
experimental arguments to mathematical arguments and deductive
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 this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
This is the second of two articles and discusses problems relating
to the curvature of space, shortest distances on surfaces,
triangulations of surfaces and representation by graphs.
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?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
What are the missing numbers in the pyramids?
Which set of numbers that add to 10 have the largest product?
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?