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 am exactly n times my daughter's age. In m years I shall be ... How old am I?
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
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. . . .
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 cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
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.
Keep constructing triangles in the incircle of the previous triangle. What happens?
How many noughts are at the end of these giant numbers?
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
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.
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.
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.
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.
The sums of the squares of three related numbers is also a perfect square - can you explain why?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
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?
Can you rearrange the cards to make a series of correct mathematical statements?
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.
When is it impossible to make number sandwiches?
What is the largest number of intersection points that a triangle and a quadrilateral can have?
This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
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.
Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.
A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?
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.
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?
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.
Can you discover whether this is a fair game?
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
From a group of any 4 students in a class of 30, each has exchanged Christmas cards with the other three. Show that some students have exchanged cards with all the other students in the class. How. . . .
An article which gives an account of some properties of magic squares.
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 looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.