Transcript The World Cup

Teacher Notes:
There is a famous problem in Discrete Mathematics
called ‘The Bridges of Konigsberg’ in which it is said to
be impossible to cross each bridge in the town of
Konigsberg, in Russia, once and only once if you wish to
return to your start point (you may start anywhere).
This problem is mathematically very similar to the
problem of trying to draw a shape without taking your
pen of the page and without retracing any line you have
A Don’t take your pen off the page!
Draw the diagram of the envelope below without
going over the same line twice and without taking
your pen off the paper.
To show that you have done it you must show your
starting and finishing points clearly and mark out
your route.
Answers:
It is possible to do if you start at any corner (vertex) that
has an ODD number of lines touching it (i.e. you must
start at the bottom right or bottom left hand corner). You
will always finish at the other ‘odd vertex’.
B
Don’t take your pen off the page!
Draw the diagram of Pythagoras’ Theorem below without
going over the same line twice and without taking your pen
off the paper.
To show that you have done it you must show your starting
and finishing points clearly and mark out your route.
Answers:
The only ‘vertices’ with an odd number of ‘edges’ (lines)
coming from them are shown in red here:
So, providing you start at one of those two ‘vertices’, you
will finish at the other one.
C
Don’t take your pen off the page!
Draw the diagram of the ‘Star of David’ below
without going over the same line twice and without
taking your pen off the paper.
To show that you have done it you must show your
starting and finishing points clearly and mark out
your route.
Answers:
Because this ‘Star of David’ doesn’t have any ‘odd
vertices’ you can start and finish anywhere – it’s easy!
(Or easier anyway).
Incidentally, if you have any picture with more than two
‘odd vertices’ it is impossible to draw without taking your
pen of or retracing your steps (hence walking the
bridges of Konigsberg is impossible unless you retrace
your steps or teleport from one section to another!)