MathDB

Hexagonal pieces numbered by positive integers are placed on the cells of a hexagonal board with side

n

. Two adjacent cells are left empty, and thanks to it some pieces can be moved. Two pieces with common sides exchanged places (see an example in the attachment 2).
Prove that if

n \ge 3

the second arrangement cannot be obtained from the first one by moving piece
Note. Moving a piece a requires two adjacent empty cells. For instance, if they are on the right of a (attachment 1, left figure), a can be moved right till it touches an angle (attachment 1, middle figure), and then it can be moved upward right or downward right (attachment 1, right figure)

Problem Statement