Problem:
Bildert works in a cubicle in an office which consists of 27 cubicles arranged in a cube. Any two cubicles sharing a wall have a connecting door on this wall; for example, the corner cubicles have exactly 3 doors, while the center cubicle has 6 doors: one on each wall, one on the floor, and one on the ceiling. If Bildert starts at the central cubicle, can he visit each of the other 26 cubicles exactly once (i.e. without revisiting any cubicles)?
Solution:
Solution I. The answer is no. Denote the central cubicle by , and denote the vertex, edge and face cubicles by and , respectively. The trip must start with and include every one of the 's, 's, and 's. The sequence must begin with . Each cubicle is adjacent only to cubicles, and each cubicle except for the very first one, is adjacent only to cubicles. This means that for the remaining 's and 's that follow the initial , at least 12 new 's are needed. This means that we need at least 's, impossible.
Solution II. Divide the cubicles into two subsets depending on the parity of the sums of coordinates of each cubicle. (Thus, in the above notation, one subset consists of cubicles and 's, and the other subset consists of 's and 's.) Each move alternates between the two subsets. The starting subset has 13 cubicles, while the other subset has 14 cubicles - obviously we cannot keep alternating, because we'll run short of cubicles in the starting subset. Hence, Bildert cannot visit all cubicles without repetitions.