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.

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

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. . . .

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.

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

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. . . .

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. . . .

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

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

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

This is an interactivity in which you have to sort into the correct order the steps in the proof of the formula for the sum of a geometric series.

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

Sort these mathematical propositions into a series of 8 correct statements.

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and record your findings in truth tables.

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

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

This article stems from research on the teaching of proof and offers guidance on how to move learners from focussing on experimental arguments to mathematical arguments and deductive reasoning.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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

Keep constructing triangles in the incircle of the previous triangle. What happens?

Which of these triangular jigsaws are impossible to finish?

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

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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...

Kyle and his teacher disagree about his test score - who is right?

Prove Pythagoras' Theorem using enlargements and scale factors.

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.

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.

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.

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...

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 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.

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

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

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?

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.

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.

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

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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.