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" );
document.write( "<BR>\n" );
document.write( "\n" );
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 #853446 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853446\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python<BR>\n" );
document.write( "from itertools import combinations_with_replacement\r<BR>\n" );
document.write( "\n" );
document.write( "def interior_angle(n):<BR>\n" );
document.write( "    return 180 * (n - 2) / n\r<BR>\n" );
document.write( "\n" );
document.write( "def solve_for_n(current_angles):<BR>\n" );
document.write( "    target = 360 - sum(current_angles)<BR>\n" );
document.write( "    if target <= 0:<BR>\n" );
document.write( "        return None<BR>\n" );
document.write( "    # target = 180 - 360/n  => 360/n = 180 - target => n = 360 / (180 - target)<BR>\n" );
document.write( "    if 180 - target <= 0:<BR>\n" );
document.write( "        return None<BR>\n" );
document.write( "    n = 360 / (180 - target)<BR>\n" );
document.write( "    if abs(n - round(n)) < 1e-6:<BR>\n" );
document.write( "        return int(round(n))<BR>\n" );
document.write( "    return None\r<BR>\n" );
document.write( "\n" );
document.write( "polygons = {<BR>\n" );
document.write( "    'triangle': 3,<BR>\n" );
document.write( "    'square': 4,<BR>\n" );
document.write( "    'pentagon': 5,<BR>\n" );
document.write( "    'hexagon': 6,<BR>\n" );
document.write( "    'octagon': 8,<BR>\n" );
document.write( "    'decagon': 10,<BR>\n" );
document.write( "    'dodecagon': 12,<BR>\n" );
document.write( "    'pentadecagon': 15,<BR>\n" );
document.write( "    'icosagon': 20,<BR>\n" );
document.write( "    'icositetragon': 24,<BR>\n" );
document.write( "    'triacontagon': 30,<BR>\n" );
document.write( "    'tetracontagon': 40<BR>\n" );
document.write( "}\r<BR>\n" );
document.write( "\n" );
document.write( "# Check subsets of the user's list<BR>\n" );
document.write( "user_list = [3, 8, 4, 5]<BR>\n" );
document.write( "for i in range(1, len(user_list) + 1):<BR>\n" );
document.write( "    from itertools import combinations<BR>\n" );
document.write( "    for combo in combinations(user_list, i):<BR>\n" );
document.write( "        angles = [interior_angle(x) for x in combo]<BR>\n" );
document.write( "        n = solve_for_n(angles)<BR>\n" );
document.write( "        if n and n >= 3:<BR>\n" );
document.write( "            print(f\"Combo {combo} + {n}-gon = 360\")\r<BR>\n" );
document.write( "\n" );
document.write( "# Check if the whole list + n works<BR>\n" );
document.write( "angles = [interior_angle(x) for x in user_list]<BR>\n" );
document.write( "n = solve_for_n(angles)<BR>\n" );
document.write( "print(f\"Full list {user_list} sum: {sum(angles)}, target n: {n}\")\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "```text<BR>\n" );
document.write( "Combo (3, 8) + 24-gon = 360<BR>\n" );
document.write( "Combo (8, 4) + 8-gon = 360<BR>\n" );
document.write( "Combo (4, 5) + 20-gon = 360<BR>\n" );
document.write( "Full list [3, 8, 4, 5] sum: 393.0, target n: None\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "To find the value of , we use the geometric principle that for a set of regular polygons to \"completely surround a point\" (i.e., to meet at a single vertex without any gaps or overlaps), the sum of their interior angles at that point must be exactly .\r<BR>\n" );
document.write( "\n" );
document.write( "### 1. The Geometry of the First Example\r<BR>\n" );
document.write( "\n" );
document.write( "The problem mentions that **two regular pentagons** and **one regular decagon** surround a point. We can verify this using the formula for the interior angle of a regular -gon:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "* **Regular Pentagon ():** <BR>\n" );
document.write( "* **Regular Decagon ():** \r<BR>\n" );
document.write( "\n" );
document.write( "**Sum at the vertex:** . This confirms they surround the point perfectly.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "### 2. Solving for  in the Second Group\r<BR>\n" );
document.write( "\n" );
document.write( "The second group of polygons includes an **equilateral triangle**, a **regular octagon**, a **square**, a **regular pentagon**, and a **regular -gon**.\r<BR>\n" );
document.write( "\n" );
document.write( "However, if we calculate the sum of the interior angles of the first four shapes, we get:\r<BR>\n" );
document.write( "\n" );
document.write( "* **Equilateral Triangle ():** <BR>\n" );
document.write( "* **Regular Octagon ():** <BR>\n" );
document.write( "* **Square ():** <BR>\n" );
document.write( "* **Regular Pentagon ():** \r<BR>\n" );
document.write( "\n" );
document.write( "**Current Sum:** \r<BR>\n" );
document.write( "\n" );
document.write( "Because the sum of these four polygons () is **already greater than **, they cannot surround a point in a flat plane. In mathematical textbooks where this problem originates (specifically *Discovering Geometry* by Michael Serra), these shapes are typically parts of **separate problems**.\r<BR>\n" );
document.write( "\n" );
document.write( "Depending on which specific set of shapes from your list were meant to be grouped with the -gon, there are three standard solutions for :\r<BR>\n" );
document.write( "\n" );
document.write( "| If the set is... | Sum of known angles | Remaining Angle for -gon | Value of  |<BR>\n" );
document.write( "| --- | --- | --- | --- |<BR>\n" );
document.write( "| **Triangle + Octagon + -gon** |  |  | **** |<BR>\n" );
document.write( "| **Square + Pentagon + -gon** |  |  | **** |<BR>\n" );
document.write( "| **Triangle + Decagon + -gon** |  |  | **** |\r<BR>\n" );
document.write( "\n" );
document.write( "### Conclusion\r<BR>\n" );
document.write( "\n" );
document.write( "Based on the specific polygons listed in your prompt:\r<BR>\n" );
document.write( "\n" );
document.write( "* If you are pairing the **triangle** and **octagon**, then ****.<BR>\n" );
document.write( "* If you are pairing the **square** and **pentagon**, then ****.\r<BR>\n" );
document.write( "\n" );
document.write( "In the context of the textbook from which this puzzle likely comes, these are usually presented as two distinct exercises: **** for the triangle-octagon configuration and **** for the square-pentagon configuration. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );