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. . . .
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
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. . . .
How many noughts are at the end of these giant numbers?
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?
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.
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. . . .
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
Keep constructing triangles in the incircle of the previous triangle. What happens?
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. . . .
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.
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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.
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. . . .
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Use this interactivity to sort out the steps of the proof of the formula for the sum of an arithmetic series. The 'thermometer' will tell you how you are doing
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 tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
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?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
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.
Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.
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.
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. . . .
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.
An article which gives an account of some properties of magic squares.
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.
A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?
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.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
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?
As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Tom writes about expressing numbers as the sums of three squares.
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!
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
Can you work out where the blue-and-red brick roads end?
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Follow the hints and prove Pick's Theorem.
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.
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Can you rearrange the cards to make a series of correct mathematical statements?
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.