Easy but nice regular hexagon procedure
Source: German TST 2004, Exam V, Problem 3
May 1, 2004
vectorinvariantgeometrycomplex numberscombinatorics solvedcombinatorics
Problem Statement
We attach to the vertices of a regular hexagon the numbers , , , , , . Now, we are allowed to transform the numbers by the following rules:
(a) We can add an arbitrary integer to the numbers at two opposite vertices.
(b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle.
(c) We can subtract an integer from one of the six numbers and simultaneously add to the two neighbouring numbers.
Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with , , , , , ; can we now get , , , , , , or , , , , , , or , , , , , , or , , , , , , or , , , , , ?)