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)=ax^{4}+b√(3)x^{3}+cx^{2}+d√(3)x+e

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

(In reply to

re(3): soln by Jer)

Jer,

Thank you again. The broken "here" link was actually a typo. Fixed. "Here" was written simply as meant (i.e. I do not know any proof of why this arrangement is optimal, although the citation for Erich Friedman '99 might hold the answer.)

SL

*Edited on ***December 1, 2018, 10:49 am**