\documentclass[crop=false]{standalone}

\usepackage{tikz,xstring,fp}
\usetikzlibrary{calc,intersections}

%Set dimensions
\newcommand\secs{20}    % number of side divisions
\newcommand\Width{10}   % width of page
\newcommand\Height{10}  % height of page
\newcommand\edgecut{1}  % how many rows - 1 to cut of on all edges, like OP (1 is no rows)

\FPeval\nWidth{clip(\Width+2)}
\FPeval\nHeight{clip(\Height+2)}

\usepackage[paperwidth=\nWidth cm, paperheight=\nHeight cm,margin=.5cm]{geometry}

\begin{document}

\thispagestyle{empty}

\topskip0pt
\vspace*{\fill}

\begin{center}

\begin{tikzpicture}
%Mark side intervals
\foreach \i in {1,...,\secs}{
  \coordinate (b\i) at (\i*\Width/\secs-\Width/\secs,0);
  \coordinate (r\i) at (\Width,\i*\Height/\secs-\Height/\secs);
  \coordinate (t\i) at (\Width-\i*\Width/\secs+\Width/\secs,\Height);
  \coordinate (l\i) at (0,\Height-\i*\Height/\secs+\Height/\secs);
}
%Mark coordinates
\foreach \j in {1,...,\secs}{
  \foreach \i in {1,...,\secs}{
    \coordinate (br\i-\j) at (intersection of r\i--b1 and b\j--l1);
    \coordinate (rt\i-\j) at (intersection of t\i--r1 and r\j--b1);
    \coordinate (tl\i-\j) at (intersection of l\i--t1 and t\j--r1);
    \coordinate (lb\i-\j) at (intersection of b\i--l1 and l\j--t1);
  }
}
%Fill in sections
\FPeval\nsecs{clip(\secs-1)}
\foreach \i in {\edgecut,...,\nsecs}{
  \FPeval\resulti{clip(\i / 2)}
  \FPeval\ip{clip(\i + 1)}
  \IfInteger{\resulti}{
    \fill[fill=black] (br\i-\secs)--(rt\secs-\i)--(rt\secs-\ip)--(br\ip-\secs);
    \fill[fill=black] (rt\i-\secs)--(tl\secs-\i)--(tl\secs-\ip)--(rt\ip-\secs);
    \fill[fill=black] (tl\i-\secs)--(lb\secs-\i)--(lb\secs-\ip)--(tl\ip-\secs);
    \fill[fill=black] (lb\i-\secs)--(br\secs-\i)--(br\secs-\ip)--(lb\ip-\secs);
  }{}
  \foreach \j in {\edgecut,...,\nsecs}{
    \FPeval\resultj{clip(\j / 2)}
    \FPeval\jp{clip(\j + 1)}
    \IfInteger{\resulti}{
      \IfInteger{\resultj}{
        \foreach \k in {br,rt,tl,lb}{
          \fill[fill=black] (\k\i-\j)--(\k\i-\jp)--(\k\ip-\jp)--(\k\ip-\j);
        }
      }{}
    }{
      \IfInteger{\resultj}{}{
        \foreach \k in {br,rt,tl,lb}{
          \fill[fill=black] (\k\i-\j)--(\k\i-\jp)--(\k\ip-\jp)--(\k\ip-\j);
        }      
      }
    }
  }
}
%Center box
\fill[fill=black] (br\secs-\secs)--(rt\secs-\secs)--(tl\secs-\secs)--(lb\secs-\secs);

%To draw whole grid:
%\foreach \i in {1,...,\secs}{
%  \draw (b1)--(r\i) (r1)--(t\i) (t1)--(l\i) (l1)--(b\i);
%}

%To draw just bounding box:
\draw (0,0)--(\Width,0)--(\Width,\Height)--(0,\Height)--(0,0);

\end{tikzpicture}

\end{center}

\vspace*{\fill}

\end{document}