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?
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 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. . . .
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.
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.
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.
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?
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.
An inequality involving integrals of squares of functions.
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
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Here the diagram says it all. Can you find the diagram?
When if ever do you get the right answer if you add two fractions
by adding the numerators and adding the denominators?
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
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.
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.
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...
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
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.
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. . . .
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.
By considering powers of (1+x), show that the sum of the squares of
the binomial coefficients from 0 to n is 2nCn
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.
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?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Explore a number pattern which has the same symmetries in different bases.
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?
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.
These proofs are wrong. Can you see why?
Have a go at being mathematically negative, by negating these
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
To find the integral of a polynomial, evaluate it at some special
points and add multiples of these values.
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?
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?
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.
Which of these roads will satisfy a Munchkin builder?
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. . . .
Follow the hints and prove Pick's Theorem.
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?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
An introduction to some beautiful results of Number Theory
Can you invert the logic to prove these statements?