All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
More Than 22! (Posted on 2005-08-18) Difficulty: 3 of 5
Let T(n) be a triangular portion of the triangular grid with n points on a side. It is an unsolved problem what the maximum number of points is that can be selected from T(n) so that no three selected points are the vertices of an equilateral triangle. For small values of n the maximum appears to be 2n-2. However, for T(12), shown below, I found more than 2*12-2=22 points, no three of which are the vertices of an equilateral triangle! How many can you find?
o
o o
o o o
o o o o
o o o o o
o o o o o o
o o o o o o o
o o o o o o o o
o o o o o o o o o
o o o o o o o o o o
o o o o o o o o o o o
o o o o o o o o o o o o
Examples:

Here's an example of 5 points selected from T(4), no three of which are the vertices of an equilateral triangle.
s
o o
o s s
o s o s
It turns out that this selection of 5 cannot be increased to 6 without three of the selected points being the vertices of an equilateral triangle. If we add the first point in the second row, we get
s
s o
o s s
o s o s
Notice that three of the 6 s's are the vertices of an equilateral triangle.

There is a better selection of 6 points in T(4) no three of which are the vertices of an equilateral triangle.

See The Solution Submitted by McWorter    
Rating: 4.1667 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): Solution | Comment 25 of 42 |
(In reply to re: Solution by Bob Smith)

Yes, it seems Tristan has found a sweet spot as to where to place the upper triangle. Bob, both of your examples have equi-tri's right in the middle. For the 25-tri, the equi-tri (5-sided) is almost a perfect dialtion of the big triangle right in the middle, while in the 26 case, the killers are rotated a bit. Do you see them?
  Posted by owl on 2005-08-19 16:18:40

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (1)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information