Map Coloring Puzzles 169442. THE FOUR-COLOR MAP THEOREM
For just about fifty years various mathematicians, including De Morgan,
Cayley, Kempe, Heawood, Heffter, Wernicke, Birkhoff, Franklin, and many
others have attempted to prove the truth of this theorem, and in a long and
learned article in the American Mathematical Monthly for July-August, 1923,Professor Brahana, of the University of Illinois, states that "the problem is
still unsolved." It is simply this, that in coloring any map under the condition
that no contiguous countries shall be colored alike, not more than four colors
can ever be necessary. Countries only touching at a point, like two Blues and
two Yellows at a in the diagram, are not contiguous. If the boundary line ca
had been, instead, at cb, then the two Yellows would be contiguous, but that
would simply be a different map, and I should have only to substitute Green
for that outside Yellow to make it all right. In fact, that Yellow might have
been Green as the map at present stands.
I will give, in condensed form, a suggested proof of my own which several
good mathematicians to whom I have shown it accept as quite valid. Two
others, for whose opinion I have great respect, think it fails for a reason that
the former maintain will not "hold water." The proof is in a form that any-
body can understand. It should be remembered that it is one thing to be con-
vinced, as everybody is, that the thing is true, but quite another to give
a rigid proof of it.