MathDB

Let

n \ge 2

be a positive integer. A grasshopper is moving along the sides of an

n \times n

square net, which is divided on

n^2

unit squares. It moves so that а) in every

1 \times 1

unit square of the net, it passes only through one side b) when it passes one side of

1 \times1

unit square of the net, it jumps on a vertex on another arbitrary

1 \times 1

unit square of the net, which does not have a side on which the grasshopper moved along. The grasshopper moves until the conditions can be fulfilled. What is the shortest and the longest path that the grasshopper can go through if it moves according to the condition of the problem? Calculate its length and draw it on the net.

Problem Statement