A polite number can be written as the sum of two or more
consecutive positive integers. Find the consecutive sums giving the
polite numbers 544 and 424. What characterizes impolite numbers?
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. . . .
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.
Tom writes about expressing numbers as the sums of three squares.
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.
An article which gives an account of some properties of magic squares.
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.
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.
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.
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?
Here the diagram says it all. Can you find the diagram?
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
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.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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.
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.
Find the smallest positive integer N such that N/2 is a perfect
cube, N/3 is a perfect fifth power and N/5 is a perfect seventh
Follow the hints and prove Pick's Theorem.
The nth term of a sequence is given by the formula n^3 + 11n . Find
the first four terms of the sequence given by this formula and the
first term of the sequence which is bigger than one million. . . .
What can you say about the common difference of an AP where every term is prime?
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.
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?
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
Take a complicated fraction with the product of five quartics top
and bottom and reduce this to a whole number. This is a numerical
example involving some clever algebra.
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.
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 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. . . .
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
Have a go at being mathematically negative, by negating these
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.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing
Can you invert the logic to prove these statements?
The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .
Explore a number pattern which has the same symmetries in different bases.
Can you explain why a sequence of operations always gives you perfect squares?
These proofs are wrong. Can you see why?
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.
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
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. . . .
An introduction to some beautiful results of Number Theory
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. . . .
Sort these mathematical propositions into a series of 8 correct
Can you rearrange the cards to make a series of correct mathematical statements?
Can you work through these direct proofs, using our interactive