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23. 11. 2024.
Sweet Division Problems: From Chocolate Bars to Honeycomb Strips
Abstract We consider finite portions of the regular hexagonal lattice and count the ways of dividing narrow strips of such a lattice into a given number of parts. We prove that such divisions are enumerated by the odd-indexed Fibonacci numbers, thus providing a new combinatorial interpretation of that sequence. We offer three different proofs of this result. Consequently, we obtain a new combinatorial proof of a well-known Fibonacci-related identity. At the end of the paper, we interpret our results in the context of graph compositions and indicate some possible directions for further research.