Henry Ernest Dudeney/Puzzles and Curious Problems/214 - Colouring the Map/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $214$

Colonel Crackham asked his young son one morning to colour all the $26$ districts in this map

in such a way that no two contiguous districts should be the same colour.

The lad looked at it for a moment, and replied,

"I haven't enough colours by one in my box."

This was found to be correct.

How many colours had he?

He was not allowed to use black or white -- only colours.

The solver will be excused for giving the perfectly reasonable answer of $3$.

However, the tricksy answer given by Dudeney is:

Two!

It is immediately apparent by inspection of the general central area that at least $4$ colours are needed.

By the Four Color Theorem it is known that no more than $4$ are needed.

Hence, being one short of $4$, there are only $3$ colours in the box.

Dudeney's reasoning is that if he has two colours in his box, he can create a third colour by mixing the two, but cannot make a fourth.

Hence and so.

$\blacksquare$