\^^M{ \frametitle{Conclusion: Rebuilding with Linear Algebra} To solve a tiling problem, we look for an underlying grid of cells that define both the region and the tiles. {\it{This isn't always possible!}} \vskip 0.1in \begin{itemize} \item{Equations: Each region cell must be covered, just once.} \item{Equations: Each tile must be used, just once.} \item{Variables: Each rotated, reflected, translated tile remaining in region} \item{Equations + Variables: {\bf{underdetermined linear system}} $Ax=b$.} \item{ {\bf{Reduced Row Echelon Form}} lets us analyze the system.} \item{ {\bf{Linear Programming Software}} solves big systems.} \item{We seek {\bf{binary}} vectors $x$ whose entries are only 0 or 1.} \item{ There may be no such solutions at all.} \item{If there are {\bf{free variables}}, we may have multiple solutions.} \end{itemize} \vskip 0.1in Any solution $x$ tells us exactly how to use the pieces so we can put a broken object back together... }