  We consider the tiling of an $n$-board (a $1\times n$ array of square cells
of unit width) with half-squares ($\frac12\times1$ tiles) and
$(\frac12,\frac12)$-fence tiles. A $(\frac12,\frac12)$-fence tile is composed
of two half-squares separated by a gap of width $\frac12$. We show that the
number of ways to tile an $n$-board using these types of tiles equals
$F_{n+1}^2$ where $F_n$ is the $n$th Fibonacci number. We use these tilings to
devise combinatorial proofs of identities relating the Fibonacci numbers
squared to one another and to other number sequences. Some of these identities
appear to be new.
