An article which gives an account of some properties of magic squares.
When you think of spies and secret agents, you probably wouldn’t think of mathematics. Some of the most famous code breakers in history have been mathematicians.
This short article gives an outline of the origins of Morse code and its inventor and how the frequency of letters is reflected in the code they were given.
Published December 2008,September 2008,December 2011,February 2011.
The possibility that any map could be coloured so that adjacent
areas, however they are combined, needs only four colours, is very
simple to describe. However, we have seen how extremely difficult
it was to prove. In the accompanying article, I have highlighted
the essential stages needed before in any proof can be successful,
be it in high-level mathematics, or in the classroom. It is also
essential to realize that a proof may not last forever. As happened
here, different areas of mathematics may influence the way we look
at a problem, and precise definition of the things we are dealing
with is often quite difficult.
The following notes offer some suggestions for the classroom,
and indicate that some of the most apparently abstract ideas can
lead to important practical applications.
Euler's Formula for polyhedra appears in may school textbooks,
but often merely as an exercise in filling in a table for the five
regular solids. But what, exactly, is a polyhedron? There is plenty
of opportunity to show different objects (packaging of various
kinds, objects with curved surfaces, cones, pyramids joined at the
apex, etc.) to see if they fit the formula, and if they do not,
whether we want to call them 'polyhedra', (or even change the
conditions for the formula).
Euler is also well-known for the 'Bridges of Konigsberg'
problem, but few take the opportunity of linking this with Hamiltonian
paths , the Icosian Game and Graph Theory. See http://en.wikipedia.org/wiki/Seven_Bridges_of_Konigsberg
Cauchy's original conception which led to his proof of the
polyhedron formula was to imagine cutting out a face of the
polyhedron and squashing it flat onto the plane, thus obtaining the
formula $f + v - e = 1$ (see the example with the cube). It is
relatively easy to obtain a picture of a tetrahedron in this way,
if you look down on it from above. However, making the maps of
other objects is more difficult.
A projection like this forms a map . It is not the same as the
net of the
Cauchy's proof started by triangulating the faces of the
'squashed' polyhedron and then removing the edges one by one, thus
preserving the formula $f + v - e = 1$. However, there are
problems. If you triangulate the dodecahedron, and begin to remove
the edges, what problems do you encounter, and what can you do
about them? Try this with any map (not necessarily representing a
polyhedron) does it work?
It is relatively easy to find the conditions for two-colouring
and three-colouring. Here is a simple 'straight line' map. How many
colours are needed here? What about the surrounding 'sea'? Do we
include it or not?
Try other maps - for example imagine the spokes on a wheel. Can
you colour the spaces between the spokes with two colours, or will
you need three? The examples given of Cayley's discoveries could
Pupils could use these ideas not just for challenging each
other, but also to give the conditions why or why not a map is
Four-colouring does not work in three dimensions. Frederic
Guthrie showed that if the 'countries' were flexible coloured rods,
they could be laid to touch each other as often as we like, so that
many more colours would be needed. You could think of the 'warp and
weft' in weaving or in basket making, where the same strands will
touch each other in many places.
However, maps drawn on surfaces which are not topologically
equivalent to a sphere, pose more difficult problems. For example,
it has been shown that at least seven colours are needed for a map
drawn on a torus.
A Hamiltonian path (circuit or cycle) is a path that visits each
vertex of the graph once and only once (except for the vertex which
is the start and finish). The path does not have to travel along
every edge to complete the circuit.
Note that the graphs shown in the puzzle on the NRICH link are
all 'cubic' graphs (each vertex is a meeting point for three
edges). Are there any Hamiltonian paths on graphs that are not
In the 1850s Thomas Kirkman (1805-1895) first studied circuits
on polyhedra, and Hamilton produced "The Icosian Game" in 1857 (see
Robin Wilson's Four Colours Suffice pages 108-112).
In the 1930s mathematicians began to apply the study of circuits
on graphs to the 'Travelling Salesman Problem' to find the most
efficient routes for delivering goods to a number of different
Since then, Graph Theory has become a large area of mathematics
with important applications for example in traffic networks, and
information flow, biology, atomic structure, and the design of
You can find out more about snarks, and more examples of them
The idea of a dual in geometry first appeared in the work of
Pascal (1623-1662) and Desargues (1591-1661) in the seventeenth
century. A good example of the idea in plane geometry is where the
pairs of opposite sides of a hexagon can be extended to meet in
three points which lie on the same straight line. Pascal Called
this the Mysterious Hexagram Theorem. An interactive version of the
dual of Pascal's Theorem by Brianchon (1783-1864) can be found at:
Duals of simple polyhedra are not difficult to visualise. By
putting a point in each of the faces of a tetrahedron, and joining
the points with lines, we get its dual. For a cube, its dual is an
Duality is an equivalence relation. The dual of a dual is
topologically equivalent to the original object. For example in the
case of polyhedra, the distances may change but the relations
between the faces, vertices and edges remain the same. The concept
of duality was used to transform the map colouring problem into
colouring vertices, and finds applications for example in power
circuits, information flow, and in dealing with
Following the development of topology in the early 20th century,
mathematicians began to develop an algebra for the classification
Knots are three-dimensional objects which can be easily made
using lengths of cord (cord shoelaces are a convenient size). Just
as Tait began, pupils can make up their own knots. It can soon be
demonstrated that there are right-handed and left handed knots
which cannot be transformed one into another.
Good clear diagrams of knots can be found in the Rolfsen Knot
Table at: http://katlas.math.toronto.edu/wiki/Image:Rolfsen_240.png
Knot theory is one of the most exciting fields of study in
mathematics because of its many important applications in
chemistry, pharmacy, biology and physics.
One area in which the left and right-handedness of molecules is
very important is in the pharmaceutical industry. The drug
Thalidomide was prescribed to pregnant women in the 1960s as a
treatment for morning sickness. The drug did reduce morning
sickness, but also caused horrible birth defects. The left-handed
molecule was curing the sickness, but the right-handed molecule was
causing severe damage to the foetus. Since this disaster, drug
companies have had to be very careful about the effects of the
mirror images of drug molecules.
More recently, the algebra associated with knot theory has found
applications in particle physics. It's strange to think that
Kelvin's vortex theory for atoms has eventually come full