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 different cubes can be painted with three blue faces and
three red faces? A boy (using blue) and a girl (using red) paint
the faces of a cube in turn so that the six faces are painted. . . .
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. . . .
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
A 'doodle' is a closed intersecting curve drawn without taking
pencil from paper. Only two lines cross at each intersection or
vertex (never 3), that is the vertex points must be 'double points'
not. . . .
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. . . .
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
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 any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
If you think that mathematical proof is really clearcut and
universal then you should read this article.
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?
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.
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.
Here are some examples of 'cons', and see if you can figure out where the trick is.
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
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.
Prove that, given any three parallel lines, an equilateral triangle
always exists with one vertex on each of the three lines.
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.
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. . . .
An article which gives an account of some properties of magic squares.
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.
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. . . .
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.
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.
Eight children enter the autumn cross-country race at school. How
many possible ways could they come in at first, second and third
A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.
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.
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. . . .
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
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 jar used to hold perfumed oil. It contained enough oil to fill
granid silver bottles. Each bottle held enough to fill ozvik golden
goblets and each goblet held enough to fill vaswik crystal. . . .
We have exactly 100 coins. There are five different values of
coins. We have decided to buy a piece of computer software for
39.75. We have the correct money, not a penny more, not a penny
less! Can. . . .
Write down a three-digit number Change the order of the digits to
get a different number Find the difference between the two three
digit numbers Follow the rest of the instructions then try. . . .
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.
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
L triominoes can fit together to make larger versions of
themselves. Is every size possible to make in this way?
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
I start with a red, a blue, a green and a yellow marble. I can
trade any of my marbles for three others, one of each colour. Can I
end up with exactly two marbles of each colour?
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?
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. . . .
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?
Which of these roads will satisfy a Munchkin builder?
Can you fit Ls together to make larger versions of themselves?
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
Take any whole number q. Calculate q^2 - 1. Factorize
q^2-1 to give two factors a and b (not necessarily q+1 and q-1). Put c = a + b + 2q . Then you will find that ab+1 , bc+1 and ca+1 are all. . . .
Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then
a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if. . . .
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
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?
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!