Jump to content
BrainDen.com - Brain Teasers
  • 0
bonanova

Equilateral Triangle: Color and distance II

Question

Suppose we add a constraint similar to the one I made in answering BMAD's

 

Four colors must be used in equal measure on each side of a unit equilateral triangle.

What is the greatest distance between two points of the same color that is unavoidable?

 

Without having given much thought, I'm wondering how this answer relates to 1/2:

with added constraint will it be greater?

Share this post


Link to post
Share on other sites

2 answers to this question

Recommended Posts

  • 0

Ignore my previous post. I got my numbers mixed up.

 

With the constraint, the largest unavoidable distance between points of the same color is 7/8.

 

First, to correct my previous post, the lower bound is sqrt(3)/2, which is the height of the equilateral triangle, not 1/sqrt(2).

 

Now, to minimize the distance to the opposite vertex, we must place all the points of the same color as close to the midpoint of the opposite side as possible. So, the 1/4 length in the middle of the opposite side is painted the same color as the opposite vertex and results in the longest distance of 7/8. Now, we need to paint the rest of the triangle without exceeding that distance. 

Paint the 1/4 length from the vertex in the same color as the vertex on both sides from the vertex. Repeat for all three vertices using 3 different colors. Now the only unpainted area is 2 segments of length 1/8 on each side. The largest distance between unpainted points is 3/4. So paint that using 4th color and we're done.

post-9659-0-25620500-1427387089_thumb.pn

Share this post


Link to post
Share on other sites
  • 0

it must be greater than 1/2. In fact, it must be greater than 1/sqrt(2). A vertex must be of some color. With the constraint, the same color must appear somewhere on the opposite side, every point of which is at least 1/sqrt(2) away.

Share this post


Link to post
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Answer this question...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...

  • Recently Browsing   0 members

    No registered users viewing this page.

×
×
  • Create New...