Fractions of permutations. An application to Sudoku Roberto Fontana DIMAT Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy article info Article history: Received 15 July 2010 Received in revised form 31 May 2011 Accepted 2 June 2011 Available online 16 June 2011 Keywords: Design of experiments Permutation matrix Sudoku abstract We study how to simplify fractional factorial design generation by exploiting the a priori knowledge that can be derived from the orthogonality constraints that the fractional factorial design itself must satisfy. We work on Sudoku puzzles that can be considered as a special case of Latin squares in the class of gerechte designs. We prove that the generation of a Sudoku is equivalent to that of a fraction of a proper set of permutations. We analyse both the 4  4 and the 9  9 Sudoku types. & 2011 Elsevier B.V. All rights reserved. 1. Introduction Sudoku is a very popular game. In its most common form, the objective of the game is to complete a 9  9 grid with the digits from 1 to 9. Each digit must appear once only in each column, each row and each of the nine 3  3 boxes. It is known that Sudoku grids are special cases of Latin squares in the class of gerechte designs, see Bailey et al. (2008). In this work we investigate the possibility of reducing the complexity of Sudoku generation by considering Sudoku as the disjoint union of simpler objects, i.e., the Sudoku permutation matrices that will be defined in Section 1.1. This idea is similar to that of considering a partition of a fraction into regular fractions (Fontana and Pistone, 2010a). 1.1. Sudoku permutation matrices We refer to Dahl (2009) for the description of a Sudoku in terms of Sudoku permutation matrices. Let us consider an N  N matrix, where N ¼ n 2 and n is a positive integer. Its row and column positions (i,j) are coded with the integer from 0 to n 2 1. We define boxes B k, l , k, l ¼ 0, ... , n1 as the following sets of positions: B k, l ¼ fði, jÞ : kn ri oðk þ 1Þn, ln rj oðl þ 1Þng It follows that any N  N matrix A can be partitioned into submatrices A kl corresponding to boxes B k, l . An N  N matrix S is a Sudoku, if in each row, in each column and in each box, each of the integers 1, ... , N appears exactly once. In the Sudoku literature, the set of boxes B b, l , l ¼ 0, ... , n1 constitutes the bth band, b ¼ 0, ... , n1, while the set of boxes B k, s , k ¼ 0, ... , n1 constitutes the sth stack, s ¼ 0, ... , n1. Let us define a Sudoku permutation matrix P, referred to as an S-matrix P, as a permutation matrix of order N which has exactly one ‘‘1’’ in each submatrix P k, l corresponding to boxes B k, l , k, l ¼ 0, ... , n1. An N  N Sudoku S identifies N matrices P i , i ¼ 1, ... , N, where P i is the S-matrix corresponding to the positions occupied by the integer i. It follows that a Sudoku Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jspi Journal of Statistical Planning and Inference 0378-3758/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2011.06.001 E-mail address: roberto.fontana@polito.it Journal of Statistical Planning and Inference 141 (2011) 3697–3704