How to Construct High Barrycades
Abstract
Given two positive integers height h and order n, the barrycade construction problem asks for a set of h permutations of the integers from $1$ to n such that all the proper partial sums given by these permutations are pairwise distinct. The name barrycade was coined by Richard K. Guy and refers to Barry Cipra, who introduced this kind of arrangement problem. A simple calculation shows that a solution can only exist for n ≥ 2h−2 and it is conjectured that there always exists a solution for every height h ≥ 2 and order n ≥ 2h−2. In this work, for every height h ≥ 1, we present a construction of a barrycade of height h and order n = 2h+3. We also present a randomized heuristic that allows us to find a barrycade of height h and conjectured optimal order n=2h−2, for every height 2 ≤ h ≤ 50. Thus, we confirm the conjectured optimal order for all heights up to $50$. We also consider a related corral construction problem, where the permutations define a cyclic arrangement. In this setting, for every height h ≥ 1, we present a construction of a corral of height h and order n=2h. A heuristic approach, similar to the one used for barrycades, allows us to find a corral of height h and conjectured optimal order n=2h−1, for every height 1 ≤ h ≤ 50. We confirm Tomoki Nakamigawa's conjecture on well-dispersed partitions of cyclic groups for the number of parts up to $20$.
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BibTeX
@misc{binieda2026construct,
title = {{How to Construct High Barrycades}},
author = {Jakub Binięda and Michał Dębski and Grzegorz Gutowski and Mateusz Milewski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24373},
eprint = {2609.24373},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24373},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - How to Construct High Barrycades
AU - Jakub Binięda
AU - Michał Dębski
AU - Grzegorz Gutowski
AU - Mateusz Milewski
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24373
UR - https://arxiv.org/abs/2609.24373
AB - Given two positive integers height h and order n, the barrycade construction problem asks for a set of h permutations of the integers from $1$ to n such that all the proper partial sums given by these permutations are pairwise distinct. The name barrycade was coined by Richard K. Guy and refers to Barry Cipra, who introduced this kind of arrangement problem. A simple calculation shows that a solution can only exist for n ≥ 2h−2 and it is conjectured that there always exists a solution for every height h ≥ 2 and order n ≥ 2h−2. In this work, for every height h ≥ 1, we present a construction of a barrycade of height h and order n = 2h+3. We also present a randomized heuristic that allows us to find a barrycade of height h and conjectured optimal order n=2h−2, for every height 2 ≤ h ≤ 50. Thus, we confirm the conjectured optimal order for all heights up to $50$. We also consider a related corral construction problem, where the permutations define a cyclic arrangement. In this setting, for every height h ≥ 1, we present a construction of a corral of height h and order n=2h. A heuristic approach, similar to the one used for barrycades, allows us to find a corral of height h and conjectured optimal order n=2h−1, for every height 1 ≤ h ≤ 50. We confirm Tomoki Nakamigawa's conjecture on well-dispersed partitions of cyclic groups for the number of parts up to $20$.
ER -