We refer to a completed Sudoku puzzle as a Sudoku Square of order 3. We define a  more restricted, but natural, extension --- a Super Sudoku Square of order n, and determine for which orders n one exists

Hoffman, Dean

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Volume 1 (2020), Article 1 https://research.library.kutztown.edu/contact/Super Sudoku SquaresDean G. Hoffman∗11Department of Mathematics and Statistics, Auburn UniversityFirst submitted: July 25, 2020Accepted: October 5, 2020Published: November 20, 2020AbstractWe refer to a completed Sudoku puzzle as a Sudoku Square of order 3. We definea more restricted, but natural, extension — a Super Sudoku Square of order n, anddetermine for which orders n one exists.1 IntroductionIn recent years, sudoku puzzles have become extremely popular. The modern day sudokupuzzle first appeared in 1979 as a puzzle called “Number Place” in Dell Magazine [6]. Theywere designed by Howard Garns [2], a freelance puzzle constructor and retired architect. Asudoku puzzle is a 9×9 square grid in which every cell contains exactly one symbol (typicallydenoted with integers 1 through 9) in such a way that each 1× 9 row contains each symbolexactly once, each 9×1 column contains each symbol exactly once, and each 3×3 sub-squaretiling the grid starting at the top left (often called blocks or regions) contains each symbolexactly once. An example of a completed Sudoku is presented in Figure 1.∗Corresponded by James Hammer (jmhammer@cedarcrest.edu)1Communications on Number Theory and Combinatorial Theory 1 (2020), Article 1 29 2 4 1 6 8 5 7 38 1 6 3 5 7 4 9 27 3 5 2 4 9 6 8 16 8 1 7 3 5 2 4 95 7 3 9 2 4 1 6 84 9 2 8 1 6 3 5 73 5 7 4 9 2 8 1 62 4 9 6 8 1 7 3 51 6 8 5 7 3 9 2 4Figure 1:Generalizations of Sudoku puzzles have been popular over the years. The most closelyrelated to this topic are Retransmission Permutation Arrays. The existence of such designshave been studied by people such as Wanless and Zhang in [13] as well as Dinitz, Paterson,Stinson, and Wei in [5]. Retransmission Permutation Arrays are n× n grids used to resolveproblems in overlapping channel transmissions.There has also been a great deal of literature written on orthogonal arrays based onSudoku puzzles such as in [8], which deals with Sudoku-like arrays, codes, and orthogonality.Other Sudoku arrays have been studied by Lorch in [10]. Samurai Sudoku-based space fillingcurves have also been a variation of Sudoku-arrays that has been used to pool data frommultiple sources in [14].Other generalizations of Sudoku puzzles such as Magic Sudoku variants have been studiedin [1]. Some variations include Sudoku using partially ordered sets in [4], modular magicsquares in [11], and strongly symmetric self-orthogonal diagonal Sudoku squares in [12].Others have taken a different approach and defined (a, b)-Sudoku Latin squares and (a, b)-Sudoku Pair Latin squares for a given factor pair (a, b) as in [7].We propose yet another way to extend the idea of the traditional Sudoku puzzle.2 Main ResultsWe coordinatize this as follows. The rows are divided into three groups, called fat rows,(fr), labeled 1, 2, 3. Each fat row consists of three skinny rows, (sr), also labelled 1, 2, 3.Similarly, we have fat columns, (fc), and skinny columns, (sc), also similarly labelled. Eachof the 81 occupied cells is assigned a label of 4 coordinates, (fr, sr, fc, sc). For example, inCommunications on Number Theory and Combinatorial Theory 1 (2020), Article 1 3the above array, symbol 9 occurs in these 9 cells: (1, 1, 3, 1), (1, 2, 1, 3), (1, 3, 2, 2), (2, 1, 1, 2),(2, 2, 2, 1), (2, 3, 3, 3), (3, 1, 2, 3), (3, 2, 3, 2), (3, 3, 1, 1).To be a valid Sudoku square, each row, and each column must contain each symbolexactly once. Furthermore, each block (= intersection of a fat row and a fat column) mustcontain each symbol exactly once. Heres the definition for values other than 3.Let N be a finite non-empty set of size |N | = n. (We took N = {1, 2, 3} above.) ASudoku square of order n, SS(n), is a function f : N4 → S, where S is a set of n2 symbols,satisfying the following properties. (Here f(c) is the symbol occupying cell c.) Each of thefollowing functions is a bijection from N2 to S: for all i, j ∈ N ,(x, y)→ f(i, j, x, y),(x, y)→ f(x, y, i, j),(x, y)→ f(i, x, j, y).To simplify notation, we denote these three properties by f(i, j, , ), f( , , i, j), f(i, , j, )respectively.We leave the proof of the following to the reader:Theorem 1. There is a SS(n) for every positive integer n.The three properties above are defined by choosing two of the four blanks in f( , , , )to be replaced by underscores . But there are six such choices. Accordingly, we define aSuper Sudoku square of order n, SSS(n), to be a function f : N4 → S, satisfying, for alli, j ∈ N , all six of the propertiesf(i, j, , ),f( , , i, j),f(i, , j, ),f(i, , , j),f( , i, j, ),f( , i, , j).Do these exist? The SS(3) above is actually a SSS(3)! Let’s check the last three propertiesat i = 3, j = 2.Communications on Number Theory and Combinatorial Theory 1 (2020), Article 1 42 6 71 5 93 4 8Figure 2: f(3, , , 2)1 6 87 3 54 9 2Figure 3: f( , 3, 2, )Communications on Number Theory and Combinatorial Theory 1 (2020), Article 1 52 6 78 3 45 9 1Figure 4: f( , 3, , 2)Note that in all three cases, the nine occupied cells contain each of the nine symbolsexactly once. Those are three of the twenty-seven properties; we leave the rest to the readerto check.So for which positive integer n is there a SSS(n)? Try n = 2, and you’ll fail. So 2 is out.To investigate this question, we need an alternate definition of super sudoku squares. Forthis, we will take S to be N ×N from now on, so our symbols are ordered pairs of elementsof N . For a positive integer k, we say an nk by k array is complete if its rows consist ofthe nk vectors with k coordinates from N , each one occurring exactly once. We define Ato be an n4 by 6 Super Sudoku array of order n, SSA(n), if its six columns are indexed bythe set {fr, sr, fc, sc, s, t}, and the n4 by 4 subarray consisting of the columns of A indexedby S is complete, for each of the 7 following sets S: {fr, sr, fc, sc}, {fr, sr, s, t}, {fc, sc, s, t},{fr, fc, s, t}, {fr, sc, s, t}, {sr, fc, s, t}, {sr, sc, s, t}.Theorem 2. There is a SSS(n) if and only if there is a SSA(n).Proof. First, suppose there is a SSS(n). We construct a SSA(n) as follows: for all (i, j, k, l) ∈N4, if cell (i, j, k, l) contains symbol (a, b), then (i, j, k, l, a, b) is a row of the SSA. Thecondition imposed by the first set S listed above states that each cell of the SSA containsexactly one symbol. The other six sets S correspond to the six conditions listed above inthe definition of SSS(n). For example, S = {fr, sc, s, t} corresponds to condition f(i, , , j).Thus an SSS(n) produces an SSA(n). But the construction is reversible, so also an SSA(n)produces an SSS(n).We’ll need some background before proceeding; see [9] for more detail. A Latin squareof order n, LS(n), is an array, with rows and columns indexed by N , filled by symbols fromCommunications on Number Theory and Combinatorial Theory 1 (2020), Article 1 6N , so that each row and each column contains each symbol exactly once. Thus a SS(n) is aLS(n2). Two such squares, A and B, are said to be orthogonal, if for all (i, j) ∈ N2, there isa unique pair (x, y) ∈ N2 so that cell (x, y) contains symbol i in square A, and symbol j insquare B. Such a pair is known as POLS(n). Here is an example for n = 3.2 3 11 2 33 1 23 2 12 1 31 3 2Figure 5:Mutually orthogonal Latin squares were introduced by the great 18th century Swissmathematician Leonhard Euler. He produced POLS(n) for all n odd, and all n a multiple of4. He observed that there is none with n = 2, and conjectured the same for any n congruentto 2 modulo 4. Nearly two centuries later, Bose, Shrikhande and Parker [3], proved there isPOLS(n) if and only if n is not 2 or 6.Here is an alternate definition. An n2 by 4 orthogonal array, which we here abbreviateto OA(n), is an array with 4 columns indexed by C = {r, c, s, t}, with the property that then2 by 2 subarray indexed by S is complete, for every two-element subset S of C.Theorem 3. There are POLS(n) if and only if there is an OA(n).Proof. Here’s the correspondence: if (x, y, i, j) is a row of the OA(n), fill cell (x, y) withsymbol i in the first square, and symbol j in the second square. We leave the details to thereader.As an example, the POLS(3) in Figure 5 gives the following OA(3):3 3 1 13 2 3 23 1 2 32 3 3 32 2 2 12 1 1 21 3 2 21 2 1 31 1 3 1Figure 6:Theorem 4. If there is a SSS(n), then there are POLS(n).Proof. For an SSA(n) A, let @ ∈ N , and define an n2 by 4 array B as follows: for each row(w, x, y, z,@,@) of A, (w, x, y, z) is a row of B. It is straightforward to show B is an OA(n),which in turn yields POLS(n).Communications on Number Theory and Combinatorial Theory 1 (2020), Article 1 7As an example, the SSS(3) in Figure 1, gives the OA(3) in Figure 6, upon taking (@,@)to be symbol 9.While a SSA(n) contains 6n4 symbols, and POLS(n) only 4n2 symbols, a constructionin the other direction seems unlikely. And yet:Theorem 5. If there are POLS(n), then there is a SSS(n).Proof. We take N to be Zn, the group of integers modulo n. Let B be POLS(n). Weconstruct a SSA(n) A as follows: for each row (w, x, y, z) of B, and each a, b ∈ N , take(w, x, y + a, z + b, a, b) to be a row of A.We must show that the n4 subarray of A with columns indexed by S is complete, for eachof the seven subsets S listed above in the definition of SSA(n). The seven verifications areall similar; we content ourselves with the proof for S = {fr, sc, s, t}. We need to show that ifi, j, k, l ∈ N , then there is a unique row (w, x, y, z) of B, and also a unique pair (a, b) ∈ N2,so that (w, x, y + a, z + b, a, b) = (i, p, q, j, k, l), for some p, q ∈ N . So a must equal k, andb must equal l. Also, we must have w = i, and z + b = j, whence z = jb = jl. But B isPOLS(n), so there are unique x, y ∈ N so that (w, x, y, z) = (i, x, y, jl) is a row of B.Note 6. Of the fifteen four-element subsets S of {fr, sr, fc, sc, s, t}, the corresponding sub-array of A is complete for thirteen of them, not just the seven required above.Corollary 7. If n is a positive integer, there is a SSS(n) if and only if n is not 2 or 6.Proof. This follows from the Bose-Shrikhande-Parker result cited above.3 AcknowledgmentsI would like to thank Dr. James Hammer for his help in the preparation of this paper forsubmission and publication.References[1] E. Arnold, R. Field, J. Lorch, S. Lucas, and L. Taalman, Nest graphs and minimalcomplete symmetry groups for magic Sudoku variants, Rocky Mountain J. Math. 45(2015), 887–901.[2] R. A. Bailey, P. J. Cameron, and R. Connelly, Sudoku, gerechte designs, resolutions,affine space, spreads, reguli, and Hamming codes, Amer. Math. Monthly 115 (2008),383–404.[3] R. C. Bose, S. S. Shrikhande, and E. T. Parker, Further results on the construction ofmutually orthogonal Latin squares and the falsity of Euler’s conjecture, Canadian J.Math. 12 (1960), 189–203.[4] A. Burgers, S. Smith, and K. Varga, Analysis of a Sudoku variation using partiallyordered sets and equivalence relations, Involve 7 (2014), 187–204.Communications on Number Theory and Combinatorial Theory 1 (2020), Article 1 8[5] J. H. Dinitz, M. B. Paterson, D. R. Stinson, and R. Wei, Constructions for retransmis-sion permutation arrays, Des. Codes Cryptogr. 65 (2012), 325–351.[6] H. Garns, Number place, Dell Pencil Puzzles and Word Games (1975), 16:6.[7] J. Hammer and D. Hoffman, Factor-pair Latin squares, Australasian. J. Comb. 69(2017), 41–57.[8] M. Huggan, G. L. Mullen, B. Stevens, and D. Thomson, Sudoku-like arrays, codes andorthogonality, Des. Codes Cryptogr. 82 (2017), 675–693.[9] C. C. Lindner and C. A. Rodger, Design Theory (2009), Chapman & Hall/CRC, Chapter6.[10] J. Lorch, Constructing ordered orthogonal arrays via Sudoku, J. Algebra Appl. 15(2016), 1650139.[11] J. Lorch and E. Weld, Modular magic sudoku, Involve 5 (2012), 173–186.[12] X. Y. Shao, Y. Zhang, and C. M. Wang, Existence of a family of strongly symmetric self-orthogonal diagonal Sudoku squares, Appl. Math. J. Chinese Univ. Ser. A 30 (2015),469–475.[13] I. M. Wanless and X. Zhang, On the existence of retransmission permutation arrays,Discrete Appl. Math. 161 (2013), 2772–2777.[14] X. Xu, P. Z. G. Qian, and Q. Liu, Samurai Sudoku-based space-filling designs for datapooling, Amer. Statist. 70 (2016), 1–8.

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