Konigsburg Bridge
Problem
• The Konigsberg
Bridge problem is
a famous
mathematical
problem studied
by many students
in geometry.
• Its citizens
pondered for a long
time whether it
was possible to
walk about the city
in such a way that
you cross all seven
bridges (yellow in
diagram) exactly
once.
• Euler trimmed the
problem down to its
basics.
• The various islands and
pieces of land became
dots
• The paths between the
pieces of land became
line segments
connecting the dots
• Can you start
at one of the
pieces of land
and travel all
seven bridges
exactly once?
• Euler proved that it
was impossible to
make such a walk.
• How did he do it?
• Let’s explore
various networks to
see if we can see
that Euler
discovered?
• First let’s define
an even point and
odd point.
An odd point
An even point
• Next you will see a series of networks.
Study they in several ways.
– Can you travel the networks? Record your
answer.
– How many odd points make up the network?
– How many even points make up the network?
– How many total points make up the network?
• Complete the chart.
Some Questions to
Ponder
• Of the networks that could be traveled,
how many of the networks had an odd
number of odd points?
• Of the networks that could be traveled,
how many of the networks had an even
number of odd points?
• Of the networks that could be traveled,
how many of the networks had an odd
number of even points?
• Of the networks that could be
traveled, how many of the networks
had an even number of even points?
• Shade in the columns on the chart
for all networks that can be
traveled.
• Look to see how many odd points a
network must have to be traveled.
Does it seem to matter how many
even points a network contains?
• What do you notice about the number
of odd points on a network that can
be traveled? (How many odd points
must it have?)
Euler’s Observation
What must be true about the
number of odd points in a network
if it can be traveled?
Using Euler’s Conjecture
• Study Konigsberg
Bridge diagram
Notice the drawing
has 4 odd points.
Use your
conjecture to add
a path so the
number of odd
points will make it
possible to travel.
Describe how this
addition fits your
conjecture.
Showing your
Understanding
Make two new networks up: one that cannot be traveled and
one that can be traveled. Use your conjecture to explain why
each can or cannot be traveled
A NETWORK THAT
CAN BE TRAVELED
A NETWORK THAT
CANNOT BE
TRAVELED