A Sudoku grid is a constrained Latin square. In this paper a reduced Sudoku grid is described, the properties of which differ, through necessity, from that of a reduced Latin square. The Sudoku symmetry group is presented and applied to determine a mathematical relationship between the number of reduced Sudoku grids and the total number of Sudoku grids for any size. This relationship simplifies...

Much has been written about the construction of sets of mutually orthogonal latin squares (MOLS). In [8], a lengthy survey of these constructions is given. Existence of MOLS is tabulated in [1], historical information appears in [9, 21], and proofs of many of the existence results appear in [1, 4, 21]. Rather than repeat these surveys here, we instead explore how some of the available construct...

A critical set in an n× n array is a set C of given entries, such that there exists a unique extension of C to an n× n Latin square and no proper subset of C has this property. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n) ≤ n2 − n. Here we show that lcs(n) ≤ n2 − 3n+ 3.

The Evans Conjecture states that a partial Latin square of order n with at most n− 1 entries can be completed. In this paper we generalize the Evans Conjecture by showing that a partial r-multi Latin square of order nwith at most n−1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Häggkvist. © 2007 Elsevier B.V. All rights reserved.

A latin square of order n is an n×n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries such that no two entries share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutually orthogon...

An N2 resolvable latin squares is a latin square with no 2×2 subsquares that also has an orthogonal mate. In this paper we show that N2 resolvable latin squares exist for all orders n with n 6= 2, 4, 6, 8

The sudoku completion problem is a special case of the latin square completion problem andboth problems are known to beNP-complete. However, in the case of a rectangular hole pattern – i.e. each column (or row) is either full or empty of symbols – it is known that the latin square completion problem can be solved in polynomial time. Conversely, we prove in this paper that the same rectangular h...

We present a ( 2 3 − o(1))-approximation algorithm for the partial latin square extension (PLSE) problem. This improves the current best bound of 1− 1 e due to Gomes, Regis, and Shmoys [5]. We also show that PLSE is APX-hard. We then consider two new and natural variants of PLSE. In the first, there is an added restriction that at most k colors are to be used in the extension; for this problem,...

A magic square of order n consists of the numbers 1 to n placed such that the sum of each row, column and principal diagonal equals the magic sum n(n +1)/2. In addition, an odd ordered magic square is associative or self-complementary if diagonally opposite elements have the same sum (n +1)/2. The magic square is said to be regular Greco-Latin if it can be decomposed as a sum of a pair of Latin...

A k-plex in a Latin square of order n is a selection of kn entries in which each row, column, and symbol is represented precisely k times.A transversal of aLatin square corresponds to the case k = 1. We show that for all even n > 2 there exists a Latin square of order n which has no k-plex for any odd k < n4 but does have a k-plex for every other k ≤ 1 2n. © 2008 Wiley Periodicals, Inc. J Combi...