  A \emph{palindrome} is a word that reads the same forwards and backwards. A
\emph{block palindrome factorization} (or \emph{BP-factorization}) is a
factorization of a word into blocks that becomes palindrome if each identical
block is replaced by a distinct symbol. We call the number of blocks in a
BP-factorization the \emph{width} of the BP-factorization. The \emph{largest
BP-factorization} of a word $w$ is the BP-factorization of $w$ with the maximum
width. We study words with certain BP-factorizations. First, we give a
recurrence for the number of length-$n$ words with largest BP-factorization of
width $t$. Second, we show that the expected width of the largest
BP-factorization of a word tends to a constant. Third, we give some results on
another extremal variation of BP-factorization, the \emph{smallest
BP-factorization}. A \emph{border} of a word $w$ is a non-empty word that is
both a proper prefix and suffix of $w$. Finally, we conclude by showing a
connection between words with a unique border and words whose smallest and
largest BP-factorizations coincide.
