) Find the Number of vertices if edges are 27 and faces 12
Answers
Unfortunately no. You can do this by trial and error on Figure 3. But it will help to alter Figure 3 slightly in the following way. Put a dot on every land mass and join two dots by a line for each bridge that connects them. So we get Figure A1 below. And then it would help to think a little more deeply than trial and error.
Konigsberg reduced to dots and lines
Figure A1: Königsberg reduced to dots and lines
Suppose that you could do the round trip walk. And suppose that you started at land mass A. Then you would have to go out from A on a bridge. So you start by using one bridge. At any time later that you came back to A you would use one bridge going in to A and one edge going out. So far you have used an odd number of bridges. Eventually you'll go back to A and use one final bridge to make the bridge count even. BUT! No land mass is attached to an even number of bridges. SO! There is no round trip walk.
Question 2
Apply the argument of Q1 to show that every land mass, except for the initial and final one, has to be attached to an even number of bridges. So can you see why the answer here is `no' too?
Question 3
It turns out that, over time, Königsberg in Prussia has become Kaliningrad, Russia. In the process of time, the seven bridges have been reduced to five. If the current position of the bridges is as shown in Figure A2, then the walk (A,B,C,D,B,A) can be done.
Konigsberg reduced to dots and lines
Figure A2: A walk with five bridges
How did you go with six bridges? Can you only find a walk that starts on one part of the city and ends up somewhere else? Why?
Question 4