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?

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. . . .

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

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.

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

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.

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. . . .

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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

When is it impossible to make number sandwiches?

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

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!

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.

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

What is the largest number of intersection points that a triangle and a quadrilateral can have?

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

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

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

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. . . .

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.)

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 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. . . .

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

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.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

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?

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

How many noughts are at the end of these giant numbers?

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

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 frogs started jumping randomly over any adjacent frog. Is it possible for them to finish up in the same order they started?

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Can you rearrange the cards to make a series of correct mathematical statements?

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

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

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry