This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .
Here the diagram says it all. Can you find the diagram?
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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.
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. . . .
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Can you work through these direct proofs, using our interactive
Can you rearrange the cards to make a series of correct mathematical statements?
You have twelve weights, one of which is different from the rest.
Using just 3 weighings, can you identify which weight is the odd
one out, and whether it is heavier or lighter than the rest?
Take a number, add its digits then multiply the digits together,
then multiply these two results. If you get the same number it is
an SP number.
An article which gives an account of some properties of magic squares.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Follow the hints and prove Pick's Theorem.
Find all positive integers a and b for which the two equations:
x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
An introduction to some beautiful results of Number Theory
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .
Given that a, b and c are natural numbers show that if sqrt a+sqrt
b is rational then it is a natural number. Extend this to 3
Can you work out where the blue-and-red brick roads end?
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
Can you explain why a sequence of operations always gives you perfect squares?
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
Janine noticed, while studying some cube numbers, that if you take
three consecutive whole numbers and multiply them together and then
add the middle number of the three, you get the middle number. . . .
Have a go at being mathematically negative, by negating these
These proofs are wrong. Can you see why?
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
Which of these triangular jigsaws are impossible to finish?
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
Some diagrammatic 'proofs' of algebraic identities and
Tom writes about expressing numbers as the sums of three squares.
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
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. . . .
Solve this famous unsolved problem and win a prize. Take a positive
integer N. If even, divide by 2; if odd, multiply by 3 and add 1.
Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...
By considering powers of (1+x), show that the sum of the squares of
the binomial coefficients from 0 to n is 2nCn
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.
Peter Zimmerman, a Year 13 student at Mill Hill County High School
in Barnet, London wrote this account of modulus arithmetic.
If you think that mathematical proof is really clearcut and
universal then you should read this article.
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
Which of these roads will satisfy a Munchkin builder?
Prove that you cannot form a Magic W with a total of 12 or less or
with a with a total of 18 or more.