I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
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. . . .
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?
This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
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?
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. . . .
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. . . .
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
Can you use the diagram to prove the AM-GM inequality?
Can you rearrange the cards to make a series of correct mathematical statements?
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.
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
Can you discover whether this is a fair game?
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 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 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. . . .
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?
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
What is the largest number of intersection points that a triangle and a quadrilateral can have?
When is it impossible to make number sandwiches?
An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
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.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
If you think that mathematical proof is really clearcut and universal then you should read this article.
A huge wheel is rolling past your window. What do you see?
How many noughts are at the end of these giant numbers?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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?
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.
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. . . .
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
Keep constructing triangles in the incircle of the previous triangle. What happens?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
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...
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.
I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?
Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?
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. . . .
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?
Which of these roads will satisfy a Munchkin builder?
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!