  Uniqueness of solution is proved for any ptychographic scheme with a random
masks under a minimum overlap condition and local geometric convergence
analysis is given for the alternating projection (AP) and Douglas-Rachford (DR)
algorithms. DR is shown to possess a unique fixed point in the object domain
and for AP a simple criterion for distinguishing the true solution among
possibly many fixed points is given.
  A minimalist scheme is proposed where the adjacent masks overlap 50\% of area
and each pixel of the object is illuminated by exactly four times during the
whole measurement process. Such a scheme is conveniently parametrized by the
number $q$ of shifted masks in each direction. The lower bound $1-C/q^2$ is
proved for the geometric convergence rate of the minimalist scheme, predicting
a poor performance with large $q$ which is confirmed by numerical experiments.
  Extensive numerical experiments are performed to explore what the general
features of a well-performing mask are like, what the best-performing values of
$q$ for a given mask are, how robust the minimalist scheme is with respect to
measurement noise and what the significant factors affecting the noise
stability are.
