Paint the tablecloth

You obviously didn't read the thread.

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That's a wonderful solution.

quasi

.

Triangles fail at their apexes with only three colors.

Without mixing to garner an additional one or two more colors (or even three), it cannot be done on a single, flat surface, and when you do do it (hahaha), you use hexes.

PT wrote: ) You have a large surface, and 3 pens, colored ) red, blue, and green, with vey fine tips. ) ) Your mission, Jim, should you decide to accept, ) is to paint the surface, such that no ) 2 points, 1 cm apart, are the same color.

Does leaving the surface uncoloured (white) also count as a colour ?

SaSW, Willem

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#EOT

Still, 4 colors might not be enough. As noted in previously-given link , "the minimum number of colors required to color the plane such that no two points at distance one from each other have the same color [...] is unknown, but has been narrowed down to one of the numbers 4, 5, 6 or 7."

The figure for the above article exhibits a 7-coloring and a sticks diagram that rules out the possibility of 3 colors being enough. Apparently no diagram is known, at present, that rules out 4 colors being enough.

The sticks diagram should not have been superimposed on the

7-coloring, but should instead stand alone, with enough of it bearing 3 colors to make it obvious that it cannot be 3-colored. The thrust of the diagram is more or less equivalent to Mike Terry's rotating triangulations argument, but probably easier to follow and far easier to formalize.

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jiw

Which triangles fail? How do they fail? Your claim makes no sense without further explanation.

You haven't proved it.

If you analyze Mike Terry's cool proof, you'll see that there are some issues that you are missing.

Even with more colors, you haven't shown that "hexes" are necessary.

quasi

And to think it was done for decades (from the '50s) on color CRTs before Sony developed their inline electron gun.

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James Waldby might have writ, in news:in7tcb$e2f$2 @dont-email.me:

FWIW, I like Mr. Terry's proof better (though it is argumably more complicated). The Wikipedia proof needs to construct a counterexample; Terry's proof just shows it *must* exist. If a different but related problem were presented, I wonder if Mr. Terry's proof method might be a better inspiration of how to proceed.

BTW, the Wikipedia article has

Since we're already cross-posted to sci.math, can someone please help elucidate this? :-)

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James Dow Allen

Triangles are a non-starter. They will not form an isolating pattern, no matter how configured.

RL

Note that this is part (a) of Problem A-4 on the 1988 Putnam exam. See, for example,