Euler proved that only two odd points can be drawn in one stroke. The existing n odd points must be drawn with n/2 strokes. Only two odd points can be drawn in one stroke, and three will have repeated routes. So n points are n/2 pairs.

Every edge of the singularity is connected, and even there are points left. All the remaining even points and a pair of singularities form a single stroke. The remaining (n/2)- 1 pen connects the remaining n-2 singularities. N must be an even number. A graph consists of odd points and even points. The number of sides of each point is called the degree. The sum of the degrees of all points is twice the number of edges, and it is even. Because each edge will be added twice when the number of times is added. Obviously, the sum of the times of all even points is even, so the sum of the times of odd points is even, which means there can only be even and odd points.