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

Home > Shapes > Geometry
5 circles in a hexagon. (Posted on 2018-11-25) Difficulty: 5 of 5
Pack five unit circles in the smallest regular hexagon possible.

The exact solution for the smallest possible side length of the hexagon is given by the largest real root of a fourth degree polynomial:

P(x)=ax4+b√(3)x3+cx2+d√(3)x+e

Where a,b,c,d,e are integers. Find them.

No Solution Yet Submitted by Jer    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
soln | Comment 5 of 9 |

While the centers of the circles do not make a regular pentagon, they do make an interesting pentagon. It is one with bilateral symmetry across the horizontal bisector of the leftmost circle, with the remaining centers forming an isosceles trapezoid. 

I believe a bona fide mathematical proof here would have to show that this is the optimal arrangement. I simply adopted it.  

I have put an annotated version of the figure here for reference and wrote my work out longhand here.  (The algebra got a bit gory, likely due to the method and coordinate system I chose.)

The bottom line is:

-63 x^4 + 219 sqrt(3) x^3 - 729 x^2 + 286 sqrt(3) x - 64 = 0 has roots:

x = (0.16660, 1.3586, 1.4964, 2.9994) with the last root being our man. 

---------------------------------------------------------------------------

Another interesting property of this pentagon is that the line "ce", extended downward, is tangent to the circle below. This is seen from the fact that the angle downward at c is 30 degrees, making a 1, 2, sqrt(3) triangle, using circle radii as sides. 


--------------------------------------------------------------------------

A personal opinion regarding this site: I believe problem authors should take the responsibility to check whether their problems (new and old) have been solved, and to then: post solutions, or mark solved by pointing to one or more solutions in the comments, or, if not solved, perhaps give hints. 
Having a clear indication of which problems remain unsolved will spur visitor interest and serve to revitalize this site.  


Edited on December 1, 2018, 12:52 am
  Posted by Steven Lord on 2018-11-29 11:55:22

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


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2020 by Animus Pactum Consulting. All rights reserved. Privacy Information