1. Regardless of the shape, the smallest positive integer $n$ that makes the total number of lattice points inside and on the boundary of a convex $n$-sided polygon $\geqslant n+1$ is ( )
A. 4
B. 5
C. 6
D. 7
1. B Consider the square $A B C D$ with vertices $A(0,0), B(0,1), C(1,1), D(1,0)$. There are no lattice points inside this square, so the required $n \geqslant 5$.

On the other hand, by classifying lattice points based on their parity, we can divide them into four categories: (odd, odd), (odd, even), (even, odd), (even, even). Among the five vertices of the convex pentagon, there must be two that belong to the same category, and the midpoint of these two vertices is also a lattice point, and this lattice point lies inside the convex pentagon, so the required $n \leqslant 5$.