<PRE><B>Question 1210522</B>
.
Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.

An equilateral triangle, a regular octagon, a square, a regular pentagon, and a regular n-gon, all with the same side length, 
also completely surround a point. Find n.
~~~~~~~~~~~~~~~~~~~~~~~~~~



(1) As the question is posed in the post, it is formulated in unsatisfactory form, in my view.
According to the context of the problem, the question should ask about possible pairs/combinations of n-gons.
Below is my solution for this modified formulation.


(2) The post by @CPhill has no clearly written answer.


His text is badly organized, badly structured, badly worded, and factually is unreadable. 
It is impossible to read it, impossible to understand, as well as impossible to discuss it.
So, from the point of view of a reader, the quality of the @CPhill solution and the quality of his presentation 
is below the floor level, i.e. is below zero.


Therefore, I present my solution below.



&lt;pre&gt;
My solution consists of two tables.  First table is for interior angles of n-gons for n = 3, 4, 5, 6, 8, 9, 10, 12.

Second table contains the answers in the form of possible combinations.



   T A B L E  1 :  Interior angles of several regular n-gons


n-gon, n:          3     4     5     6     8     9     10     12     15     18     20


interior angle,   60    90   108    120   135   140    144   150
  degrees



  T A B L E  2 :   T h e   a n s w e r 


Combinations:   n * angle1 + m * angle2 = 360 degrees   ( angle1 and angle2 are interior angles )

                6 * 60                  = 360             6 equilateral triangles

                3 * 60     + 2 * 90     = 360             3 eq. triangles + 2 squares  

                4 * 90                  = 360             4 squares

                2 * 108    + 1 * 144    = 360             2 reg.  pentagons and 1 reg. 10-gon

                2 * 135    + 1 * 90     = 360             2 reg. octagons + 1 square

                2 * 150    + 1 * 60     = 360             2 reg. 12-gons  + 1 eq. triangle



The solution method is, actually, trial and error by brute force, and I did it mentally.


@CPhill, actually, tries to do the same using a computer code,
but my presentation of the solution and of the answer is incomparable better than that by @CPhill (in my opinion).


I do not try to impress a reader by my knowledge of names for regular n-gons, as @CPhill does,
but simply present the logic, the solution and the answer in clear straightforward way.
&lt;/pre&gt;

Solved.



-------------------------



As I observe the @CPhill compositions every day,

this person simply does not know true commonly accepted standards of formulation to Math problems,
as well does not know true commonly accepted standards for presentations of their solutions.


Therefore, every day he constantly tries to break the existing standards and to re-establish them at the lowest level,
which corresponds to his own level in Math.


It is inacceptable attempt which must be resolutely rejected by the community.



</PRE>