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.
- Universal Number Partition
- divisibility and packing
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics