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
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. . . .
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.
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...
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
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?
If you think that mathematical proof is really clearcut and
universal then you should read this article.
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?
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
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. . . .
Here the diagram says it all. Can you find the diagram?
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.
An inequality involving integrals of squares of functions.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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.
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.
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
Some diagrammatic 'proofs' of algebraic identities and
Can you rearrange the cards to make a series of correct mathematical statements?
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?
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?
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
Can you work through these direct proofs, using our interactive
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
Can you see how this picture illustrates the formula for the sum of
the first six cube numbers?
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
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.
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.
An article which gives an account of some properties of magic squares.
To find the integral of a polynomial, evaluate it at some special
points and add multiples of these values.
These proofs are wrong. Can you see why?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Explore a number pattern which has the same symmetries in different bases.
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.
Prove that the internal angle bisectors of a triangle will never be
perpendicular to each other.
Can you invert the logic to prove these statements?
Which of these triangular jigsaws are impossible to finish?
Follow the hints and prove Pick's Theorem.
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
Prove that in every tetrahedron there is a vertex such that the
three edges meeting there have lengths which could be the sides of
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.
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. . . .
An introduction to some beautiful results of Number Theory
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 is the second article on right-angled triangles whose edge lengths are whole numbers.
The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .