?? solutions to the 8-queens problem.htm

?? Using Genetic Algorithm to solve the 8 Queens problem.
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<H1 align=center>Solutions to the 8-Queens Problem</H1>
<P>This problem is to place 8 queens on the chess board so that they do not 
check each other. This problem is probably as old as the chess game itself, and 
thus its origin is not known, but it is known that Gauss studied this problem. 
If we want to find a single solution, it is not difficult as shown below. If we 
want to find all possible solutions, the problem is difficult and the backtrack 
method is the only known method. For 8-queen, we have 92 solutions. If we 
exclude symmetry, there are 12 solutions. </P>
<P>Consider the general case of the n-Queens Problem </P>
<P>If n is a prime number, a solution is easily found by drawing a straight line 
in the (n, n) finite plane. Since no two straight lines can intersect at two 
points, a straight line y=ax+b where a is not equal to 1 or -1 can give a 
solution. Coordinates start from 0. </P>
<TABLE cellSpacing=0 width=255 border=0>
  <TBODY>
  <TR>
    <TD vAlign=center colSpan=9>
      <P>Example 1: n = 7, y = 2x</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%" height=18><FONT size=2>
      <P>6 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff height=18>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000 height=18>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff height=18>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000 height=18><B><FONT 
      color=#ffffff size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff height=18>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000 height=18>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff height=18>
      <P></P></TD>
    <TD vAlign=center width="20%" height=18 rowSpan=8>
      <P></P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>5 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000><B><FONT color=#ffffff 
      size=2>
      <P align=center>X</B></FONT></P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>4 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff><B><FONT size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>3 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff><B><FONT size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>2 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000><B><FONT color=#ffffff 
      size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%" height=15><FONT size=2>
      <P>1 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000 height=15>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff height=15>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000 height=15>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff height=15>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000 height=15><B><FONT 
      color=#ffffff size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff height=15>
      <P></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000 height=15>
      <P></P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>0 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff><B><FONT size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%">
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>0 </FONT></P></TD>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>1 </FONT></P></TD>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>2 </FONT></P></TD>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>3 </FONT></P></TD>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>4 </FONT></P></TD>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>5 </FONT></P></TD>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>6 </FONT></P></TD></TR>
  <TR>
    <TD vAlign=center colSpan=9>
      <P>&nbsp;</P>
      <P>n = 7, y = 3x + 1</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>6 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff><B><FONT size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="20%" rowSpan=8>
      <P>&nbsp;</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>5 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000><B><FONT color=#ffffff 
      size=2>
      <P align=center>X</B></FONT></P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>4 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000><B><FONT color=#ffffff 
      size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>3 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff><B><FONT size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>2 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000><B><FONT color=#ffffff 
      size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>1 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#000000><B><FONT color=#ffffff 
      size=2>
      <P align=center>X</B></FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD></TR>
  <TR>
    <TD vAlign=center width="10%"><FONT size=2>
      <P>0 </FONT></P></TD>
    <TD vAlign=center width="10%" bgColor=#ffffff>
      <P>&nbsp;</P></TD>
    <TD vAlign=center width="10%" bgColor=#000000>
      <P>&nbsp;</P></TD>
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