document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209329>Question 1209329</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Not including the identity transformation, the eleven transformations that preserve the regular hexagon shown are counterclockwise rotations by $60^\circ,$ $120^\circ,$ $180^\circ,$ $240^\circ$ and $300^\circ$ and reflections across the six dashed lines shown. Kristina randomly picks six transformations $T_1,$ $T_2,$ $T_3,$ $T_4$, $T_5$ and $T_6,$ with replacement, from this set of eleven. She performs these six transformations on the hexagon, in succession. The probability that the point $P$ is transformed to each of the hexagon's six vertices exactly once during this process is $\dfrac{k}{11^6}.$ What is the value of $k\,?$ <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848383 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</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=848383\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Not including the identity transformation, the eleven transformations that preserve <BR>\n" );
document.write( "the regular hexagon shown are counterclockwise rotations by 60°, 120°, 180°, 240° and 300° <BR>\n" );
document.write( "and reflections across the six dashed lines shown. Kristina randomly picks six transformations <BR>\n" );
document.write( "T_1, T_2, T_3, T_4, T_5 and T_6, with replacement, from this set of eleven. She performs <BR>\n" );
document.write( "these six transformations on the hexagon, in succession. The probability that the point P <BR>\n" );
document.write( "is transformed to each of the hexagon's six vertices exactly once during this process is k/11^6}. <BR>\n" );
document.write( "What is the value of k ? <BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In my view, this problem, as formulated in the post, &nbsp;is &nbsp;FATALLY &nbsp;DEFECTIVE &nbsp;and is not a mathematical problem, &nbsp;at all.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I will point only two defects in its formulation, &nbsp;but they just make it totally invalid.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First my objection is that \"the point &nbsp;$P$\" &nbsp;is not defined in the post, &nbsp;despite of using an article \"the\".\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Second objection is that the form &nbsp;k/11^6 &nbsp;can represent &nbsp;ANY &nbsp;number and does not represent any special/specific number,\r<BR>\n" );
document.write( "\n" );
document.write( "so, &nbsp;mathematically, &nbsp;this condition that &nbsp;\" the probability . . . is k/11^6 \" &nbsp;does not determine a concrete number\r<BR>\n" );
document.write( "\n" );
document.write( "and, &nbsp;HENCE, &nbsp;defines &nbsp;NOTHING &nbsp;and is &nbsp;EMPTY.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I have many other objectives against this formulation, &nbsp;but these two are just enough to kill this &nbsp;\" problem \" &nbsp;to the death.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The solution provided to this problem by &nbsp;AI &nbsp;is soup of words without any relevant interior logic in it.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The impression after reading this &nbsp;\" solution \" &nbsp;is as if the system tries to sell nonsense<BR>\n" );
document.write( "under the guise of &nbsp;Artificial &nbsp;Intelligence.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I made similar notices many times, &nbsp;so you just should have understanding that the current version <BR>\n" );
document.write( "of your &nbsp;AI &nbsp;is not able to recognize false &nbsp;Math problems and to distinct them from proper &nbsp;Math problems.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So, when a false/(fatally defective) &nbsp;Math problem comes, &nbsp;the system tries to treat <BR>\n" );
document.write( "it as a proper &nbsp;Math problem, &nbsp;instead of rejecting it.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From my view, it is the weakest point of the current version of your AI.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );