document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210522>Question 1210522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.\r<BR>\n" );
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document.write( "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. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853459 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53646)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853459\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Two regular pentagons and a regular decagon, all with the same side length, can completely surround a point, as shown.\r<BR>\n" );
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document.write( "An equilateral triangle, a regular octagon, a square, a regular pentagon, and a regular n-gon, all with the same side length, <BR>\n" );
document.write( "also completely surround a point. Find n.<BR>\n" );
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document.write( "(1) As the question is posed in the post, it is formulated in unsatisfactory form, in my view.<BR>\n" );
document.write( "According to the context of the problem, the question should ask about possible pairs/combinations of n-gons.<BR>\n" );
document.write( "Below is my solution for this modified formulation.\r<BR>\n" );
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document.write( "(2) The post by @CPhill has no clearly written answer.\r<BR>\n" );
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document.write( "His text is badly organized, badly structured, badly worded, and factually is unreadable. <BR>\n" );
document.write( "It is impossible to read it, impossible to understand, as well as impossible to discuss it.<BR>\n" );
document.write( "So, from the point of view of a reader, the quality of the @CPhill solution and the quality of his presentation <BR>\n" );
document.write( "is below the floor level, i.e. is below zero.\r<BR>\n" );
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document.write( "Therefore, I present my solution below.\r<BR>\n" );
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document.write( "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.\r\n" );
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document.write( "Second table contains the answers in the form of possible combinations.\r\n" );
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document.write( "   T A B L E  1 :  Interior angles of several regular n-gons\r\n" );
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document.write( "n-gon, n:          3     4     5     6     8     9     10     12     15     18     20\r\n" );
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document.write( "interior angle,   60    90   108    120   135   140    144   150\r\n" );
document.write( "  degrees\r\n" );
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document.write( "  T A B L E  2 :   T h e   a n s w e r \r\n" );
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document.write( "Combinations:   n * angle1 + m * angle2 = 360 degrees   ( angle1 and angle2 are interior angles )\r\n" );
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document.write( "                6 * 60                  = 360             6 equilateral triangles\r\n" );
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document.write( "                3 * 60     + 2 * 90     = 360             3 eq. triangles + 2 squares  \r\n" );
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document.write( "                4 * 90                  = 360             4 squares\r\n" );
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document.write( "                2 * 108    + 1 * 144    = 360             2 reg.  pentagons and 1 reg. 10-gon\r\n" );
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document.write( "                2 * 135    + 1 * 90     = 360             2 reg. octagons + 1 square\r\n" );
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document.write( "                2 * 150    + 1 * 60     = 360             2 reg. 12-gons  + 1 eq. triangle\r\n" );
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document.write( "The solution method is, actually, trial and error by brute force, and I did it mentally.\r\n" );
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document.write( "@CPhill, actually, tries to do the same using a computer code,\r\n" );
document.write( "but my presentation of the solution and of the answer is incomparable better than that by @CPhill (in my opinion).\r\n" );
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document.write( "I do not try to impress a reader by my knowledge of names for regular n-gons, as @CPhill does,\r\n" );
document.write( "but simply present the logic, the solution and the answer in clear straightforward way.\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "As I observe the @CPhill compositions every day,\r<BR>\n" );
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document.write( "this person simply does not know true commonly accepted standards of formulation to Math problems,<BR>\n" );
document.write( "as well does not know true commonly accepted standards for presentations of their solutions.\r<BR>\n" );
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document.write( "Therefore, every day he constantly tries to break the existing standards and to re-establish them at the lowest level,<BR>\n" );
document.write( "which corresponds to his own level in Math.\r<BR>\n" );
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document.write( "It is inacceptable attempt which must be resolutely rejected by the community.\r<BR>\n" );
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