$31 \cdot 21$ The number of integer solutions to the equation $(x+1)^{2}+(y-2)^{2}=1$ is
(A) 1.
(B) 2.
(C) 4.
(D) Infinite.
(China Beijing Junior High School Mathematics Competition, 1990)
[Solution] From the given, we know that $0 \leqslant x+1 \leqslant 1$, and $0 \leqslant y-2 \leqslant 1$.
Since the equation has integer solutions, we have
$\left\{\begin{array}{l}x+1=0, \\ y-2= \pm 1 ;\end{array}\right.$ or $\quad\left\{\begin{array}{l}x+1= \pm 1, \\ y-2=0 .\end{array}\right.$

Therefore, the correct choice is $(C)$.