Welcome to the staging ground for new communities! Each proposal has a description in the "Descriptions" category and a body of questions and answers in "Incubator Q&A". You can ask questions (and get answers, we hope!) right away, and start new proposals.
Are you here to participate in a specific proposal? Click on the proposal tag (with the dark outline) to see only posts about that proposal and not all of the others that are in progress. Tags are at the bottom of each post.
Comments on Can you avoid monochromatic triangles?
Parent
Can you avoid monochromatic triangles? Question
In the diagram below we have pentagon ABCDE with 5 edges and 5 diagonals. These 10 line segments are currently all colored black.
The diagram contains several triangles but for this puzzle we are only interested in the 10 triangles where all three of its vertices are vertices of pentagon ABCDE.
A triangle is considered monochromatic if all three of its edges are the same color. Currently all 10 triangles are monochromatic black.
Is it possible to recolor some of the 10 line segments red, leaving the other line segments black, so that none of the 10 triangles are monochromatic?
If yes, provide a diagram showing the recoloring.
If no, prove it is impossible.
Post
Yes.
One answer is the five exterior line segments one color and the five interior line segments another.
The reasoning is that all 10 of the triangles you care about have both exterior and interior segments. Therefore, with exterior and interior segments different colors, all of the 10 triangles must be not monochromatic.
There are possibly other solutions. I didn't look into that.
0 comment threads