I have not dealt with problems of this type:
Three integers $a$, $b$, and $c$ are written on a blackboard. Then one of them is erased and replaced by the sum of the other two diminished by 1. This operation is repeated a finite number of times until we have the final result, which is $17$, $\,1967$, and $1983$.
Could $a,b,c$ have been $(2,2,2)$ i.e. starting from $2$, $2$, $2$, can we reach $17$, $\,1967$, and $1983$?