# Signatura of magic and Latin integer squares: isentropic clans and indexing

Ian Cameron; Adam Rogers; Peter D. Loly

Discussiones Mathematicae Probability and Statistics (2013)

- Volume: 33, Issue: 1-2, page 121-149
- ISSN: 1509-9423

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topIan Cameron, Adam Rogers, and Peter D. Loly. "Signatura of magic and Latin integer squares: isentropic clans and indexing." Discussiones Mathematicae Probability and Statistics 33.1-2 (2013): 121-149. <http://eudml.org/doc/270890>.

@article{IanCameron2013,

abstract = {
The 2010 study of the Shannon entropy of order nine Sudoku and Latin square matrices by Newton and DeSalvo [Proc. Roy. Soc. A 2010] is extended to natural magic and Latin squares up to order nine. We demonstrate that decimal and integer measures of the Singular Value sets, here named SV clans, are a powerful way of comparing different integer squares.
Several complete sets of magic and Latin squares are included, including the order eight Franklin subset which is of direct relevance to magic square line patterns on chess boards. While early examples suggested that lower rank specimens had lower entropy, sufficient data is presented to show that some full rank cases with low entropy possess a set of singular values separating into a dominant group with the remainder much weaker. An effective rank measure helps understand these issues.
We also introduce a new measure for integer squares based on the sum of the fourth powers of the singular values which appears to give a useful method of indexing both Latin and magic squares. This can be used to begin cataloging a "library" of magical squares.
Based on a video presentation in celebration of George Styan's 75th at LINSTAT2012 and IWMS-21 on 19 July, 2012 at Będlewo, Poland.
},

author = {Ian Cameron, Adam Rogers, Peter D. Loly},

journal = {Discussiones Mathematicae Probability and Statistics},

keywords = {Shannon entropy; magic square; Latin square; singular value decomposition; singular value clan},

language = {eng},

number = {1-2},

pages = {121-149},

title = {Signatura of magic and Latin integer squares: isentropic clans and indexing},

url = {http://eudml.org/doc/270890},

volume = {33},

year = {2013},

}

TY - JOUR

AU - Ian Cameron

AU - Adam Rogers

AU - Peter D. Loly

TI - Signatura of magic and Latin integer squares: isentropic clans and indexing

JO - Discussiones Mathematicae Probability and Statistics

PY - 2013

VL - 33

IS - 1-2

SP - 121

EP - 149

AB -
The 2010 study of the Shannon entropy of order nine Sudoku and Latin square matrices by Newton and DeSalvo [Proc. Roy. Soc. A 2010] is extended to natural magic and Latin squares up to order nine. We demonstrate that decimal and integer measures of the Singular Value sets, here named SV clans, are a powerful way of comparing different integer squares.
Several complete sets of magic and Latin squares are included, including the order eight Franklin subset which is of direct relevance to magic square line patterns on chess boards. While early examples suggested that lower rank specimens had lower entropy, sufficient data is presented to show that some full rank cases with low entropy possess a set of singular values separating into a dominant group with the remainder much weaker. An effective rank measure helps understand these issues.
We also introduce a new measure for integer squares based on the sum of the fourth powers of the singular values which appears to give a useful method of indexing both Latin and magic squares. This can be used to begin cataloging a "library" of magical squares.
Based on a video presentation in celebration of George Styan's 75th at LINSTAT2012 and IWMS-21 on 19 July, 2012 at Będlewo, Poland.

LA - eng

KW - Shannon entropy; magic square; Latin square; singular value decomposition; singular value clan

UR - http://eudml.org/doc/270890

ER -

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