  We investigate the construction of $\pm1$-valued completely multiplicative
functions that take the value $+1$ at at most $k$ consecutive integers, which
we call length-$k$ functions. We introduce a way to extend the length based on
the idea of the "rotation trick" and such an extension can be quantified by the
number of modified primes. Under the assumption of Elliott's conjecture, this
method allows us to construct length-$k$ functions systematically for $k\geq 4$
which generalizes the work of I. Schur for $k = 2$ and R. Hudson for $k =3$.
