There are a lot of warnings, and there is one first syntax error at:

Scalar found where operator expected at ./BP_Chap11_Exer1.pl line 30, 
+near "]$j"
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Searching for that string points out this line:

              print $chessboard[$i]$j];
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The next syntax error is

syntax error at ./BP_Chap11_Exer1.pl line 19, near "$_}"
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Searching for that string points to this line:

     $chessboard[0][$_] = "W" . $back[$_};  # White Back Row
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The line numbers don't match up with your code. Are you sure that you show us the code that produces these error messages?!