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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Symmetric skew balanced starters and complete balanced Howell rotations


Authors: Ding Zhu Du and F. K. Hwang
Journal: Trans. Amer. Math. Soc. 271 (1982), 409-413
MSC: Primary 05B15; Secondary 05B10, 90D12
MathSciNet review: 654840
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Abstract: Symmetric skew balanced starters on $ n$ elements have been previously constructed for $ n = 4k + 3$ a prime power and $ 8k + 5$ a prime power. In this paper we give an approach for the general case $ n = {2^m}k + 1$ a prime power with $ k$ odd. In particular we show how this approach works for $ m = 2$ and $ 3$. Furthermore, we prove that for $ n$ of the general form and $ k > 9 \cdot {2^{3m}}$, then a symmetric skew balanced starter always exists. It is known that a symmetric skew balanced starter on $ n$ elements, $ n$ odd, can be used to construct complete balanced Howell rotations (balanced Room squares) for $ n$ players and $ 2(n + 1)$ players, and in the case that $ n$ is congruent to $ 3$ modulo $ 4$, also for $ n + 1$ players.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0654840-8
Article copyright: © Copyright 1982 American Mathematical Society