362 Answersspaces A, B, C, D, E to mere points, and representing the bridges that
connect these spaces by lines or roads. This transformation does not affect
the conditions, for there are 16 bridges or roads in one case, and 16 roads or
lines in the other, and they connect with A, B, C, D, E in precisely the same
way. It will be seen that 9 bridges or roads connect with the outside. Obvi-
ously we are free to join these up in pairs in any way we choose, provided the
roads do not cross one another. The simplest way is shown in Figure 3, where
on coming out from A, B, C, or E, we immediately return to the same point
by the adjacent bridge, leaving one point, X, necessarily in the open. In
Figure 2 there are 4 odd nodes, A, B, D, and X (if we decide on the exits and
entrances, as in Figure 3), so, as I have already explained, we require 2
strokes (half of 4) to go over all the roads, proving a perfect solution to be
impossible.
Now, let us cancel the line AB. Then A and B become even nodes, but we
must begin and end at the odd nodes, D and X. Follow the line in Figure 3,
and you will see that this can be done, omitting the line from A to B. This
route the reader will easily transform into Figure 4 if he says to himself, "Go
from X to D, from D to E, from E to the outside and return into E," and so
on. The route can be varied by linking up those outside bridges differently,
by making X an outside bridge to A or B, instead of D, and by taking
the cancelled line either at AB, AD, BD, XA, XB, or XD. In Figure 5 I make
X lead to B. We still omit AB, but we must start and end at D and X. Trans-
formed in Figure 6, this will be seen to be the precise example that I gave in
stating the problem. The reader can now write out as many routes as he likes
for himself, but he will always find it necessary to omit one line or crossing.
It is thus seen how easily sometimes a little cunning, like that of the trans-
formation shown, will settle a perplexing question of this kind.
- THE NINE BRIDGES
Transform the map as follows. Reduce the four islands, A, B, C, and D, to
mere points and extend the bridges into lines, as in Figure I, and the condi-
tions are unchanged. If you link A and B for outside communication, and
also C and D, the conditions are as in Figure 2; if you link A and D,