Henry Ernest Dudeney/Modern Puzzles/140 - The Four-Colour Map Theorem

Modern Puzzles by Henry Ernest Dudeney: $140$

The Four-Colour Map Theorem

In colouring any map under the condition that no contiguous countries shall be coloured alike,

not more than four colours can ever be necessary.

Countries only touching at a point ... are not contiguous.

I will give, in condensed form, a suggested proof of my own

which several good mathematicians to whom I have shown it accept it 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 anybody can understand.

It should be remembered that it is one thing to be convinced, as everybody is, that the thing is true,

but quite another to give a rigid proof of it.

Also see

• Four Color Theorem

Historical Note

For just about $50$ 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."

Martin Gardner, in his $1968$ repackaging 536 Puzzles & Curious Problems, refutes Dudeney's claim to have proved it.

He goes on to advertise his own contribution in his $1966$ Martin Gardner's New Mathematical Diversions from Scientific American.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Geometrical Problems: Various Geometrical Puzzles: $140$. -- The Four-Colour Map Theorem
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Combinatorial & Topological Problems: Map Coloring Puzzles: $442$. The Four-Color Map Theorem