378 Answerscountry were contiguous with each of four that are contiguous each with each.
If such a map is possible, the theorem breaks down.
First, let us consider four countries contiguous each with each. We will use
a simple transformation and suppose every two contiguous countries to be
connected by a bridge across the boundary line. The bridge may be as long as
we like, and the countries may be reduced to mere points, without affecting
the conditions. In Figures 8 and 9 I show four countries (points) connected
by bridges (lines), each with each. The relative positions of these points
is quite immaterial, and it will be found in every possible case that one
country (point) must be unapproachable from the outside.
The proof of this is easy. If three points are connected each with each by
straight lines these points must either form a triangle or lie in a straight line.
First suppose they form a triangle, YRG, as in Figure 16. Then a fourth con-
tiguous country, B, must lie either within or without the triangle. If within, it
is obviously enclosed. Place it outside and make it contiguous with Y and G,
as shown: then B cannot be made contiguous with R without enclosing either
Y or G. Make B contiguous with Y and R: then B cannot be made contiguous
with G without enclosing either Y or R. Make B contiguous with Rand G:
then B cannot be made contiguous with Y without enclosing either R or G.
Take the second case, where RYG lie in a straight line, as in Figure 17. If
B lies within the figure it is enclosed. Place B outside and make it contiguous
with Rand G, as shown: then B cannot be made contiguous with Y without
enclosing either R or G. Make B contiguous with Rand Y: then B cannot be
made contiguous with G without enclosing either R or Y. Make B contiguous
with Y and G: then B cannot be made contiguous with R without enclosing
either Y or G.
We have thus taken every possible case and found that if three countries are
contiguous each with each a fourth country cannot be made contiguous with
all three without enclosing one country.
Figure 10 is Figure 8 before the transformation, and Figure II is the same
as Figure 9, and it will be seen at once that you cannot possibly reach R from
the outside. Therefore four countries cannot possibly be drawn so that a fifth
may be contiguous with every one of them, and consequently the fifth country
may certainly repeat the color R. And if you cannot draw five countries, it is
quite obvious that you cannot draw any greater number contiguous each
with each.