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A cube side 3 is divided into 27 unit cubes

Source: Balkan MO 2001, problem 4

April 24, 2006
geometry3D geometryanalytic geometrycombinatorics proposedcombinatorics

Problem Statement

A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube nn has moved to the position originally occupied by 27−n27-n (for each n=1,2,…,26n = 1, 2, \ldots , 26)?
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