Nearly magic rectangles

Feng Shun Chai; Rakhi Singh

doi:10.1002/jcd.21667

Nearly magic rectangles

Feng Shun Chai, Rakhi Singh

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares is magic rectangles which are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by odd nearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist.

Original language	English (US)
Pages (from-to)	562-567
Number of pages	6
Journal	Journal of Combinatorial Designs
Volume	27
Issue number	9
DOIs	https://doi.org/10.1002/jcd.21667
State	Published - Sep 2019
Externally published	Yes

Keywords

magic rectangle
magic square
row-column design

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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10.1002/jcd.21667

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title = "Nearly magic rectangles",

abstract = "Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares is magic rectangles which are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by odd nearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist.",

keywords = "magic rectangle, magic square, row-column design",

author = "Chai, {Feng Shun} and Rakhi Singh",

note = "Funding Information: National Science Foundation, Grant/ Award Numbers: DMS‐1935729, DMS–1811363; Deutsche Forschungsgemeinschaft, Grant/Award Numbers: Teilprojekt C2, SFB 823; NSF, Grant/Award Numbers: DMS–1811363, DMS–1935729 Funding Information: Much of this study was completed when both the second and third authors were visiting Academia Sinica and we gratefully acknowledge hospitality and support from Academia Sinica. The research of R.S. is supported by the Deutsche Forschungsgemeinschaft (SFB 823, Teilprojekt C2) and that of J.S. is partially supported by NSF grants (DMS–1811363 and DMS–1935729). Publisher Copyright: {\textcopyright} 2019 Wiley Periodicals, Inc.",

year = "2019",

month = sep,

doi = "10.1002/jcd.21667",

language = "English (US)",

volume = "27",

pages = "562--567",

journal = "Journal of Combinatorial Designs",

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AU - Chai, Feng Shun

AU - Singh, Rakhi

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PY - 2019/9

Y1 - 2019/9

N2 - Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares is magic rectangles which are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by odd nearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist.

AB - Magic squares have been extremely useful and popular in combinatorics and statistics. One generalization of magic squares is magic rectangles which are useful for designing experiments in statistics. A necessary and sufficient condition for the existence of magic rectangles restricts the number of rows and columns to be either both odd or both even. In this paper, we generalize magic rectangles to even by odd nearly magic rectangles. We also prove necessary and sufficient conditions for the existence of a nearly magic rectangle, and construct one for each parameter set for which they exist.

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KW - magic square

KW - row-column design

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JO - Journal of Combinatorial Designs

JF - Journal of Combinatorial Designs

IS - 9

ER -

Nearly magic rectangles

Abstract

Keywords

ASJC Scopus subject areas

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