$11 \cdot 31$ Given a convex 32-sided polygon on a square grid with a side length of 1, all of its vertices are on the grid's nodes. What is the minimum possible perimeter of this polygon?
[Solution]. Consider each side of the convex 32-gon $A_{1} A_{2} \cdots A_{32}$ as a vector, thus obtaining 32 vectors that satisfy the following conditions:
(1) The starting and ending points of each vector are on the nodes of the grid paper;
(2) Any two vectors have different directions;
(3) The sum of all vectors is the zero vector.
Conversely, if there is a set of 32 vectors that satisfy conditions (1)-(3), then they must form a convex 32-gon with nodes as vertices. Therefore, our problem is reduced to finding the minimum value of the sum of the lengths of 32 vectors that satisfy conditions (1)-(3).

We move the starting points of all vectors to a node $O$, which is the center of the square in the right figure. In the right figure, there are exactly 32 vectors, and it is easy to see that it is impossible to draw another vector with $O$ as the starting point and another node as the endpoint, such that its length is shorter than the longest vector in the figure. Therefore, the minimum value we seek is $4+4 \sqrt{2}+8 \sqrt{5}+8 \sqrt{10}+8 \sqrt{13}$.