  In this paper we find infinitely many Mazur type manifolds and corks with
shadow complexity one among the 4-manifolds constructed from contractible
special polyhedra having one true vertex by using the notion of Turaev's
shadow. We also find such manifolds among 4-manifolds constructed from Bing's
house. Our manifolds with shadow complexity one contain the Mazur manifolds
$W^{\pm }(l,k)$ which were studied by Akbulut and Kirby.
