Can you invert the logic to prove these statements?
An introduction to some beautiful results of Number Theory
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
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.
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. . . .
Can you work through these direct proofs, using our interactive
Follow the hints and prove Pick's Theorem.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
With n people anywhere in a field each shoots a water pistol at the
nearest person. In general who gets wet? What difference does it
make if n is odd or even?
Can you rearrange the cards to make a series of correct mathematical statements?
An article about the strategy for playing The Triangle Game which
appears on the NRICH site. It contains a simple lemma about
labelling a grid of equilateral triangles within a triangular
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.
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.
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. . . .
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?
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.
These proofs are wrong. Can you see why?
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...
An inequality involving integrals of squares of functions.
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. . . .
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.
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
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.
By considering powers of (1+x), show that the sum of the squares of
the binomial coefficients from 0 to n is 2nCn
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.
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Here the diagram says it all. Can you find the diagram?
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.
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
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.
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.
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
This is an interactivity in which you have to sort the steps in the
completion of the square into the correct order to prove the
formula for the solutions of quadratic equations.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
Can you explain why a sequence of operations always gives you perfect squares?
Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?
Explore a number pattern which has the same symmetries in different bases.
Can you work out where the blue-and-red brick roads end?
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
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.
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
Have a go at being mathematically negative, by negating these
Prove that in every tetrahedron there is a vertex such that the
three edges meeting there have lengths which could be the sides of
Sort these mathematical propositions into a series of 8 correct