Object : Several new map colouring problems.
Click the main board at the center to colour some circular areas red.
Adjacent areas are not allowed to have the same colour.
If no more areas can be coloured red, the system will make the next colour (blue)
available. Again you colour any areas you wish with this colour, and so on.
Depending how you select your areas, to fill the board you might need three, four, five or
or even more colours.
In the default variant, you are only allowed to use 3 cololurs,
in the next variant you have to use 4 colours and so on.
Solutions are attached.
Expert level:
Some variants have a counter attached. Positions coloured by the first colour
count 1, positions coloured by the second colour count 10 and so on.
Maximise this counter. The maximum count known is also shown at the bottom.
Can you match or even beat these results?
Variant 1-1: use only 3 colours to fill the hex board
Variant 1-2: use 4 colours to fill the hex board
Variant 1-3: use 5 colours to fill the hex board
Variant 1-4: use 6 colours to fill the hex board
Variant 1-5: use 7 colours to fill the hex board
Variant 2-1: use 6 colours to fill the smaller hex board.
Variant 2-2: use 7 colours to fill the smaller hex board.
Variant 3-1 : automatical fill of a small hex board
Variant 3-2 : automatical hex board fill
Variant 3-3 : automatical fill of a large hex board
Variant 4-1: use only 2 colours to fill the square 6x6 board
Variant 4-2: use 3 colours to fill the square 6x6 board
Variant 4-3: use 4 colours to fill the square 6x6 board
Variant 4-4: use 5 colours to fill the square 6x6 board
Variant 5-1 : automatical fill of the square 6x6 board
Variant 6-1: A special map which shows that in general we need an infinite number of colours.
Use 8 colours to fill it.
In the following we always inply that adjacent areas do not carry the same colour.
The famous 4-Colour Theorem says that one only need 4 colours to colour any map.
This game 'Maxicol' looks at the same problem from more random point of view:
How many colours do we need if the first colour is distributed randomly,
then the second colour, then the third and so on?
We will call this method a 'random-colouring' of a map.
For the hexagonal grid, we learn from variant 1-5 that - to cover even the worst case -
we need at least 7 colours.
But is this the maximum for the hex grid? No proof is known.
What is the maximum if we allow any sort of 2-dimensional map?
The author has shown that in general we need an infinite number of colours!
The proof is contained in variant 5-1. The board used there can be easily generalized
to create (for any given number n) a map which needs n colours to random-colour it. |
