Self-orthogonal Diagonal Latin Squares

Description

Latin Square, (LS): An n x n matrix in which n distinct symbols are arranged such that each symbol occurs exactly once in each row and in each column.

Diagonal Latin Square, (DLS): A Latin Square in which each symbol also occurs exactly once in each of the two main diagonals.

Orthogonal Latin Squares, (OLS). Two Latin squares of order n are orthogonal if each symbol in the first square meets each symbol in the second square exactly once when they are superposed.

Self-orthogonal Diagonal Latin Square, (SODLS): A diagonal Latin square that is orthogonal to its transpose, (Rotate 7).

Construction

MagicSquaresSODLS makes SODLS, (with symbols 0 to n-1). A magic square M is made from each SODLS Q and its transpose as, (see notes):


                 M[row][col] = n x Q[row][col] + Q[col][row] + 1

Note: associative, pandiagonal, ultramagic below refer to SODLS that make these types of magic square.

There are no pandiagonal Latin squares, (Knut Vik designs), for even orders or odd orders that are a multiple of 3. For these orders, the SODLS that make pandiagonal and ultramagic squares are only weakly pandiagonal, (SOWPLS), meaning that the sum of the n numbers in each of 2n diagonals is the same.

See: associative SSSODLS, pandiagonal Knut Vik, weakly pandiagonal SOWPLS, ultramagic PSSSODLS.

SODLS makes SODLS for small orders, SODLS9 makes order 9 SODLS, SODLS10 makes order 10 SODLS.

Odd Order

Odd, not multiple of 3

Note: Pandiagonal and ultramagic here refer to Knut Vik designs.

Pandiagonal, ultramagic, and associative are easily made:

The pandiagonal method is borrowed from Inder Taneja. The ultramagic method is based on the formula Q[r][c] = (2r-c)%n for prime orders ≥ 5 by Xu and Lu. Other associative are made by swapping rows/columns of the ultramagic.

Small associative squares are also made by a backtracking procedure. Orders 11 and 13 associative made by backtracking are included in the program.

Non-prime order squares such as 25, 35, 49, 55 are sometimes made as composites of smaller ones.

Odd, multiple of 3

Note: Pandiagonal and ultramagic here refer to weakly pandiagonal.

For order 9, an associative is made by backtracking, a pandiagonal is made based on one by Inder Taneja, and an ultramagic made by backtracking is included in the program.
An order 15 associative made by backtracking is included in the program.
An order 27 pandiagonal is made based on one by Xu and Lu.

Other methods, (extending concepts from squares by Mitsutoshi Nakamura):

x ks+1 are similar to an order 21 by Mitsutoshi Nakamura.

Some squares, such as 45, 63, 75, 81 are made as composites of smaller ones.

Doubly Even Order

Note: Pandiagonal and ultramagic here refer to weakly pandiagonal.

Small order associative squares are made by a backtracking procedure.

An order 12 associative made by backtracking is included in the program. Order 24 is made as 4 5s+4 similar to a 24 magic by Mitsutoshi Nakamura. An order 8 ultramagic and an order 16 associative made by backtracking are included for variety.

Associative squares are sometimes converted to pandiagonal squares by Planck's A-D method. See note.

Some squares, such as 16, 20, 28, 32 are made as composites of smaller ones.

Singly Even Order

Note: There are no associative or pandiagonal Latin squares of singly-even order. See proofs.

An order 10 and an order 14 magic are included in the program. The 10 was made by backtracking; the 14 is adapted from one by Inder Taneja based on Bennett, Du and Zhang; Orders 18, 22, 26 and 30 are made with a diagonal, center 4x4 similar to an order 18 by Mitsutoshi Nakamura.

Other methods, (extending concepts from Mitsutoshi Nakamura's squares):

x ks+k/2 are similar to an order 22 by Mitsutoshi Nakamura.

Some squares, such as 50, 70, 90, 98 are made as composites of smaller ones.

Examples

Here are fragments of consoles for 1 1000 and some of the thousands of variations for 1000.
Note that none 3 is because there is no SODLS of order 2, 3, or 6.

How Many

The number of SODLS for some small orders is known. See notes.

* nfr: SODLS with natural order first row, (0 1 2 3 ... ).
Francis Gaspalou calculated the number of SODLS for order 8 in 2010.
Eduard Vatutin has confirmed the numbers for orders up to 8 and added OEIS sequences, (A287761, A287762).

Here is type information for the magic squares:

Transforms

A SODLS remains a SODLS under Transform 1 and Transform 2.

REFERENCES

"Graeco-Latin square" https://en.wikipedia.org/wiki/Graeco-Latin_square

Amela, Miguel Angel    miguel.amel@gmail.com
General Pico - La Pampa - Argentina
"Universal H-IXOHOXI Magic Squares of Order Eight"
http://www.multimagie.com/AmelaUHIMS.pdf
"A Sextuply Self-Orthogonal Latin Square of Order Nine and their Magic Squares", 2014
addenda/6-SOLS 9x9.pdf

Bennett, Frank E., Beiliang Du and Hantao Zhang
"Existence of self-orthogonal diagonal Latin squares with a missing subsquare", 2003
http://www.sciencedirect.com/science/article/pii/S0012365X02004612

Cao, H. and W. Li
"Existence of strong symmetric self-orthogonal diagonal Latin squares", 2011
http://www.sciencedirect.com/science/article/pii/S0012365X11000513

Gaspalou, Francis    francis.gaspalou@wanadoo.fr
Website http://www.gaspalou.fr/magic-squares/
Francis communicated the number of order 8 SODLS to an e-mail group in July, 2010. He computed the number of SODLS with the natural order first row as 1,152, and calculated the total as 1,152 x 8! = 46,448,640.

Hedayat, A and W.T.Federer
"On The Nonexistence of Knut Vik Designs For All Even Orders", 1975
https://www.jstor.org/stable/2958957?seq=1#page_scan_tab_contents

Hedayat, A
"A complete solution to the existence and nonexistence of Knut Vik designs and orthogonal Knut Vik designs", 1977
http://www.sciencedirect.com/science/article/pii/0097316577900073

Henderson, Matthew
"On Pandiagonal Strongly Symmetric Self-Orthogonal Diagonal Latin Squares (PSSSODLS)", 2012
http://mhenderson.net/assets/research/articles/12_04_16_psssodls.pdf

Kim, Kichul and V. K. Prasanna Kumar
"Perfect Latin Squares and Parallel Array Access", 1989
http://mprc.pku.edu.cn/~liuxianhua/chn/corpus/Notes/articles/isca/1989/p372-kim/p372-kim.pdf

Kochemazov, Stepan, Eduard Vatutin, and Oleg Zaikin
"Fast Algorithm for Enumerating Diagonal Latin Squares of Small Order", 2017
https://arxiv.org/abs/1709.02599

Lu, Runming, Sheng Liu and Jian Zhang
"Searching for Doubly Self-orthogonal Latin Squares", 2011
http://link.springer.com/chapter/10.1007%2F978-3-642-23786-7_41#page-1

Nakamura, Mitsutoshi    tustim@post.nifty.jp
Website http://magcube.la.coocan.jp/magcube/en/
The SODLS squares referred to above were communicated to Inder Taneja and an e-mail group by Mitsutoshi in July, 2015.

Taneja, Inder J.    ijtaneja@gmail.com
Formerly, Professor of Mathematics, Federal University of Santa Catarina,
88040-400 Florianópolis, SC, Brazil.
"Intervally Distributed, Palindromic, Selfie Magic Squares, and Double Colored Patterns", 2015
http://rgmia.org/papers/v18/v18a127.pdf
"Intervally Distributed, Palindromic and Selfie Magic Squares: Genetic Table and Colored Pattern - Orders 11 to 20", 2015
http://rgmia.org/papers/v18/v18a140.pdf
"Intervally Distributed, Palindromic and Selfie Magic Squares - Orders 21 to 25", 2015
http://rgmia.org/papers/v18/v18a151.pdf

Vatutin, Eduard I.    evatutin@rambler.ru
Website http://evatutin.narod.ru/indexeng.html

Xu, Cheng-Xu and Zhun-Wei Lu
"Pandiagonal Magic Squares", 2005
http://link.springer.com/chapter/10.1007%2FBFb0030856#page-2

Zhang, Yong, Wen Li and Jian Guo Lei
"Existence of weakly pandiagonal orthogonal Latin squares", 2013
http://link.springer.com/article/10.1007%2Fs10114-013-2274-1