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. . . .
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. . . .
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?
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. . . .
How many noughts are at the end of these giant numbers?
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
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?
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.
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
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
A huge wheel is rolling past your window. What do you see?
I am exactly n times my daughter's age. In m years I shall be ... How old am I?
After some matches were played, most of the information in the
table containing the results of the games was accidentally deleted.
What was the score in each match played?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
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?
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
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. . . .
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
Here are some examples of 'cons', and see if you can figure out where the trick is.
Carry out cyclic permutations of nine digit numbers containing the
digits from 1 to 9 (until you get back to the first number). Prove
that whatever number you choose, they will add to the same total.
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. . . .
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.
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.
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
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.
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
Replace each letter with a digit to make this addition correct.
An article which gives an account of some properties of magic squares.
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
If you know the sizes of the angles marked with coloured dots in
this diagram which angles can you find by calculation?
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?
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?
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.
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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.
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
Which hexagons tessellate?
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
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