Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
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.
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. . . .
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 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. . . .
Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.
How many tours visit each vertex of a cube once and only once? How
many return to the starting point?
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.
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
Some simple ideas about graph theory with a discussion of a proof
of Euler's formula relating the numbers of vertces, edges and faces
of a graph.
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
Given the nets of 4 cubes with the faces coloured in 4 colours,
build a tower so that on each vertical wall no colour is repeated,
that is all 4 colours appear.
Label this plum tree graph to make it totally magic!
Find all the ways of placing the numbers 1 to 9 on a W shape, with
3 numbers on each leg, so that each set of 3 numbers has the same
The Four Colour Conjecture was first stated just over 150 years ago, and finally proved conclusively in 1976. It is an outstanding example of how old ideas can be combined with new discoveries. prove. . . .
Explore creating 'factors and multiples' graphs such that no lines joining the numbers cross
Imagine you had to plan the tour for the Olympic Torch. Is there an efficient way of choosing the shortest possible route?
How many different colours would be needed to colour these
different patterns on a torus?
in how many ways can you place the numbers 1, 2, 3 … 9 in the
nine regions of the Olympic Emblem (5 overlapping circles) so that
the amount in each ring is the same?
Show there are exactly 12 magic labellings of the Magic W using the
numbers 1 to 9. Prove that for every labelling with a magic total T
there is a corresponding labelling with a magic total 30-T.
Look for the common features in these graphs. Which graphs belong together?
This article explains the use of the idea of connectedness in networks, in two different ways, to bring into focus the basics of the game of Go, namely capture and territory.
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
The game of go has a simple mechanism. This discussion of the principle of two eyes in go has shown that the game does not depend on equally clear-cut concepts.
A personal investigation of Conway's Rational Tangles. What were
the interesting questions that needed to be asked, and where did
The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.
Given probabilities of taking paths in a graph from each node, use
matrix multiplication to find the probability of going from one
vertex to another in 2 stages, or 3, or 4 or even 100.
Find out some of the mathematics behind neural networks.
Given the graph of a supply network and the maximum capacity for
flow in each section find the maximum flow across the network.
An account of how mathematics is used in computer games including
geometry, vectors, transformations, 3D graphics, graph theory and
This article explains the concepts involved in scientific
mathematical computing. It will be very useful and interesting to
anyone interested in computer programming or mathematics.