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 frame.

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself?

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

When is it impossible to make number sandwiches?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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.

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

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?

Which of these triangular jigsaws are impossible to finish?

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. . . .

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

This is the second article on right-angled triangles whose edge lengths are whole numbers.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

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 power.

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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.

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.

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?

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

Can you explain why a sequence of operations always gives you perfect squares?

An introduction to some beautiful results of Number Theory

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

Can you work out where the blue-and-red brick roads end?

Have a go at being mathematically negative, by negating these statements.

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 variables.

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

By proving these particular identities, prove the existence of general cases.

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.