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abc cubical boxes in a x b x c rectangular stack, bee flying inside the stack

Source: Norwegian Mathematical Olympiad 2013 - Abel Competition p4b

September 4, 2019

combinatorial geometrycombinatorics

Problem Statement

A total of

a \cdot b \cdot c

cubical boxes are joined together in a

a \times b \times c

rectangular stack, where

a, b, c \ge 2

. A bee is found inside one of the boxes. It can fly from one box to another through a hole in the wall, but not through edges or corners. Also, it cannot fly outside the stack. For which triples

(a, b, c)

is it possible for the bee to fly through all of the boxes exactly once, and end up in the same box where it started?

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