Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: cotpi 69 - Black and white plane
Replies: 26   Last Post: Oct 9, 2013 10:23 AM

 Messages: [ Previous | Next ]
 Phil Carmody Posts: 2,219 Registered: 12/7/04
Re: cotpi 69 - Black and white plane
Posted: Oct 9, 2013 5:04 AM

David Bernier <david250@videotron.ca> writes:
> On 10/02/2013 10:29 PM, Mike Terry wrote:
> > "quasi" <quasi@null.set> wrote in message
> > news:v9vo49lvr7j4ns7i82s869ou749lgdmrup@4ax.com...

> >> Pubkeybreaker wrote:
> >>> Susam Pal wrote:
> >>>>
> >>>> How can you construct a plane where every point is coloured
> >>>> either black or white such that two points of the same colour
> >>>> are never a unit distance apart?
> >>>>
> >>>> -- Originally posted at:
> >>>> http://cotpi.com/p/69/
> >>>> Puzzles IRC channel:
> >>>> ##puzzles on irc.freenode.net
> >>>> Puzzles IRC webchat:
> >>>> http://webchat.freenode.net/?channels=##puzzles

> >>>
> >>> It does not seem possible. Take any point. Suppose it is
> >>> white. It is the center of a circle (of radius 1). All points
> >>> on that circle must then be black. But given any point on
> >>> that circle there is another point on the circle at distance 1.
> >>> Both points are the same color.
> >>>
> >>> Unless, of course, I am being completely stupid.

> >>
> >> No, your proof is fine.

> >
> > The question was asked here a few years ago whether the plane can be
> > coloured along the lines above but using three colours. The answer is no,
> > and the proof is similar but less obvious...

> [...]
>
> There is a web-page called "Chromatic Number of the Plane" by
> Alexander Bogomolny that briefly discusses the question of
> the minimum number of colours needed.
>
> The relevant definition, copied from there, is:
> ``The smallest number of colors needed in a coloring of the plane to
> ensure that no monochromatic pair is at the unit distance apart is
> called the chromatic number Chi of the plane."
>
> Ref.:
> < http://www.cut-the-knot.org/proofs/ChromaticNumber.shtml > .
>
> Two or three colours won't do, from which we see that Chi >= 4.
> A 7-colouring of a regular-hexagon tiling of the plane shows
> that seven colours will do, from which we see that Chi <= 7.

Follow-up questions:
1) What range(s) of edge-length for said hexagon yield a valid tiling?

2) Does Golomb's ten-vertex four-colour graph have to be rotationally
symmetric?

My brief quick stabs at answers are hidden in my headers, I welcome
corrections.

Phil
--
The list of trusted root authorities in your browser included the
United Arab Emirates-based Etisalat, which was caught secretly
http://www.wired.com/threatlevel/2009/07/blackberry-spies/

Date Subject Author
10/2/13 cotpi
10/2/13 Pubkeybreaker
10/2/13 quasi
10/2/13 Mike Terry
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 quasi
10/6/13 quasi
10/6/13 David Bernier
10/6/13 David Bernier
10/6/13 quasi
10/6/13 David Bernier
10/9/13 Phil Carmody
10/9/13 David Bernier
10/2/13 Eric Lafontaine
10/2/13 Michael F. Stemper
10/2/13 quasi
10/2/13 quasi
10/2/13 Haran Pilpel
10/2/13 quasi
10/2/13 quasi
10/2/13 Ted Schuerzinger