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

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

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.

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

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.)

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. . . .

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.

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

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

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

A introduction to how patterns can be deceiving, and what is and is not a proof.

When is it impossible to make number sandwiches?

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

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

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?

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.

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

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, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

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?

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

If you think that mathematical proof is really clearcut and universal then you should read this article.

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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

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.

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.

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

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.

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

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

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. . . .

Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.

Which of these triangular jigsaws are impossible to finish?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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?

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

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?

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

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

Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?