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
soln by Steven Lord)
Kudos! Your longhand solution looks great. It's pretty much the way I did it. The algebra is quite a slog. The first link doesn't work though.
The issue with solved/unsolved/no official solution has always been a problem with this site. Other puzzle sites require you to submit problems with a solution or a least an answer. Just because a problem has been solved doesn't mean a person can't solve it independently.
One nice thing is you can submit problems that you can't solve here. I have a problem like that in the queue right now: colored triangle.

Posted by Jer
on 20181129 15:44:58 