  Sand pile models are dynamical systems describing the evolution from $N$
stacked grains to a stable configuration. It uses local rules to depict grain
moves and iterate it until reaching a fixed configuration from which no rule
can be applied. The main interest of sand piles relies in their {\em Self
Organized Criticality} (SOC), the property that a small perturbation | adding
some sand grains | on a fixed configuration has uncontrolled consequences on
the system, involving an arbitrary number of grain fall. Physicists L. Kadanoff
{\em et al} inspire KSPM, a model presenting a sharp SOC behavior, extending
the well known {\em Sand Pile Model}. In KSPM($D$), we start from a pile of $N$
stacked grains and apply the rule: $D-1$ grains can fall from column $i$ onto
the $D-1$ adjacent columns to the right if the difference of height between
columns $i$ and $i+1$ is greater or equal to $D$. This paper develops a formal
background for the study of KSPM fixed points. This background, resumed in a
finite state word transducer, is used to provide a plain formula for fixed
points of KSPM(3).

Discrete Mathematics