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📄 solutions to the 8-queens problem.htm

📁 Using Genetic Algorithm to solve the 8 Queens problem.
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<P>We can easily find 28 solutions by a=2, 3, 4, 5, and b=0, 1, 2, 3, 4, 5, 6. 
</P>
<P>The ratio of analytical solutions for the total solutions for some small p is 
as follows: </P>
<P>p=5, 10/10, 100%</P>
<P>p=7, 28/40, 70% </P>
<P>p=11, 99/2680, 4%</P>
<P>For composite numbers n=pq, we can make a direct product of the p-queen and 
q-queen problems. That is, each queen position of the p-queen problem is 
regarded as a solution of the q-queen problem. We can change the roles of p and 
q. Thus for 35=5*7, we can generate 10*(40)^5 + 40*(10)^7 solutions. </P>
<P>To generate one solution for a general n, let the plane coordinated by i=0, 
..., n-1 and j=0, ..., n-1. </P>
<P>Suppose n is even. For any k, </P>
<P>(1) If n is not 6k+2, </P>j = 2i+1, for 0 &lt;= i &lt; n/2 <BR>j = 2i mod n, 
for n/2 &lt;= i &lt; n 
<P>Example 2. n=6 </P>
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<P>(2) If n is not 6k </P>j = (n/2 + 2i -1) mod n, for 0 &lt;= i &lt; n/2<BR>j = 
(n/2 + 2i + 2) mod n, for n/2 &lt;= i <N <P>Example 3. n=8 
<P></P>
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      size=2>
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      size=2>
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      <P>3 </FONT></P></TD>
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    <TD vAlign=top width="11%" bgColor=#000000><B><FONT color=#ffffff size=2>
      <P align=center>X</B></FONT></P></TD>
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