### Abstract

This note examines the likelihood of packing two identical one dimensional shelves of integer length L by items whose individual lengths are divisors of L, given that their combined length sums-up to 2L. We compute the number of packing failures and packing successes for integer shelve lengths L, 1 ≤ L ≤ 1000, by implementing a dynamic programming scheme using a problem specific "boundedness property". The computational results indicate that the likelihood of a packing failure is very rare. We observe that the existence of packing failures is tied to the number of divisors of L and prove that the number of divisors has to be at least 8 for a packing failure to exist.

Original language | English (US) |
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Pages (from-to) | 189-198 |

Number of pages | 10 |

Journal | Discrete Mathematics, Algorithms and Applications |

Volume | 2 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2010 |

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### Keywords

- divisibility and packing
- Universal Number Partition

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics, Algorithms and Applications*,

*2*(2), 189-198. https://doi.org/10.1142/S1793830910000565