Construction and Analysis of Samurai Sudoku

Authors: A. Danbaba

Abstract:

Samurai Sudoku consists of five Sudoku square designs each having nine treatments in each row (column or sub-block) only once such the five Sudoku designs overlaps. Two or more Samurai designs can be joint together to give an extended Samurai design. In addition, two Samurai designs, each containing five Sudoku square designs, are mutually orthogonal (Graeco). If we superimpose two Samurai designs and obtained a pair of Latin and Greek letters in each row (column or sub-block) of the five Sudoku designs only once, then we have Graeco Samurai design. In this paper, simple method of constructing Samurai designs and mutually orthogonal Samurai design are proposed. In addition, linear models and methods of data analysis for the designs are proposed.

Keywords: Samurai design, Graeco samurai design, sudoku design, row or column swap.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1123717

References:

[1] Hui-Dong, M. and Ru-Gen, X. (2008). Sudoku Square — a New Design in Field Experiment, Acta Agron Sin, 34(9), 1489–1493.
[2] Varun, S. B. (2015). Sudoku Squares as Experimental Designs, International Journal of Engineering Trends and Technology, 28(5), 229 – 235.
[3] Okagbue, H. I., Adamu M.O., Oguntunde P.E., and Opanuga A. A. (2015), Some Notes on the 3-Factor Analysis of 9×9 Sudoku, Research Journal of Applied Sciences, 10(7), 284 – 286.
[4] Michael, F. A. S. and Bikas K. S. (2014). SudoKu as an Experimental Design - Beyond the Traditional Latin Square Design, Statistics and Applications, 12 (1&2), 15-20.
[5] Kumar, A., Varghese C. Varghese E. and Jaggi S. (2015). On the construction of designs with three-way blocking. Model Assisted Statistics and Applications 10, 43–52 43.
[6] Subramani, J. and Ponnuswamy, K.N. (2009). Construction and analysis of Sudoku designs, Model Assisted Statistics and Applications, 4(4), 287-301.
[7] Subramani, J. (2012), Construction of Graeco Sudoku Square Designs of Odd Orders, Bonfring International Journal of Data Mining, 2 ( 2), 37 – 41.
[8] Lorch, J. (2009), Mutually orthogonal families of linear sudoku solutions, J. Austral. Math. Soc., 87,409-420.
[9] Subramani, J. (2013). Construction and analysis of orthogonal (Graeco) Sudoku square designs, Model Assisted Statistics and Applications, 8, 239–246.
[10] Fontana, R. (2014). Random Latin squares and Sudoku designs generation, Electronic Journal of Statistics, 8, 883–893.
[11] Xu, X., Haaland, B., and Qian, P. Z. G (2011), Sudoku-Based Space-Filling Designs,"Biometrika, 98, 711-720.
[12] Xu, X., Qian, P. Z. G, and Qing Liu (2014), Sudoku-Based Space-Filling Designs for Multi-Source Inference.
[13] Denes, J., Keedwell, A.D. (1974). Latin Squares and their Applications. London: English Universities Press limited.
[14] Hedayat, A. S., Sloame N. J. A. and Stufken J. (1999). Orthogonal Arrays: Theory and Applications, Sprinder verlag, New York.
[15] Goyeneche, D., Daniel A., Jose I. L., Arnau R., and Karol Z. (2015) Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices. Physical. Review A 92, 032316 – 15.
[16] Vanpoucke, J (2012). Mutually orthogonal Latin squares and their generalizations, A Master thesis submitted to the Faculty of Sciences, Ghent University.
[17] Donovan, D., Haaland B. and Nott D.J (2015). A Simple Approach to Constructing Quasi-Sudoku-based Sliced Space-Filling Designs. arxiv.1502.05522v1.
[18] Milovanovic, I. Z., E.R. Glogic, E. I. Milovanovic, M. P. Bekakos, M. K. Stojcev (2015), Permutation Matrices of Reverse r-th Stride, ser. A: appl. Math. Inform. And mech. 5(2), 79-84.
[19] Puntanen, S., Styan G. P. H., and Isotalo J. (2011). Matrix Tricks for Linear Statistical Models, Springer. New York.