Latin squares and superqueens

Let L be a Latin square of order n with entries from {0, 1,…, n ? 1}. In addition, L is said to have the (n, k) property if, in each right or left wrap around diagonal, the number of cells with entries smaller than k is exactly k. It is established that a necessary and sufficient condition for the existence of Latin squares having the (n, k) property is that of (2|n ? 2| k) and (3|n ? 3| k). Also, these Latin squares are related to a problem of placing nonattacking queens on a toroidal chessboard.     

Crisscross Latin squares

Let A be a Latin square of order n. Then the jth right diagonal of A is the set of n cells of A: {(i,j+i):i=0,1…,n?1(modn); and the jth left diagonal of A is the set {(i,j?i):i=0,1…,n?1(modn); A diagonal is said to be complete if every element appears in it exactly once. For n = 2m even, we introduce the concept of a crisscross Latin square which is something in between a diagonal Latin square and a Knut Vik design. A crisscross Latin square is a Latin square such that all the jth right diagonals for even j and all the jth left diagonals for odd j are complete. We show that a necessary and sufficient condition for the existence of a crisscross Latin square of order 2m is that m is even.     

On avoiding odd partial Latin squares and r-multi Latin squares

We show that for any positive integer k?4, if R is a (2k-1)×(2k-1) partial Latin square, then R is avoidable given that R contains an empty row, thus extending a theorem of Chetwynd and Rhodes. We also present the idea of avoidability in the setting of partial r-multi Latin squares, and give some partial fillings which are avoidable. In particular, we show that if R contains at most nr/2 symbols and if there is an n×n Latin square L such that δn of the symbols in L cover the filled cells in R where 0<δ<1, then R is avoidable provided r is large enough.     

Magic squares with magic inverses

Characteristic polynomials are used to determine when magic squares have magic inverses. A resulting method constructs arbitrary examples of such squares.     

An algorithm for constructing magic squares

In this paper we introduce a product operation on the set of all matrices of integers. Using this operation we give an algorithm to construct an infinite family of magic squares and show that the set of all magic squares forms a free monoid.     

Latin squares with one subsquare

We look at two classes of constructions for Latin squares which have exactly one proper subsquare. The first class includes known squares due to McLeish and to Kotzig and Turgeon, which had not previously been shown to possess unique subsquares. The second class is a new construction called the corrupted product. It uses subsquare‐free squares of orders m and n to build a Latin square of order mn whose only subsquare is one of the two initial squares. We also provide tight bounds on the size of a unique subsquare and a survey of small order examples. Finally, we foreshadow how our squares might be used to create new Latin squares devoid of proper subsquares—so called N_∞ squares. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 128–146, 2001     

Transversals of additive Latin squares

LetA={a ₁, …,a _k} andB={b ₁, …,b _k} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈S _ksuch that the sums α _i +b _i , 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method in various finite and infinite fields we extend Alon’s result to the groups (ℤ _p ) ^a and in the casek<p, and verify Snevily’s conjecture for every cyclic group of odd order. Supported by Hungarian research grants OTKA F030822 and T029759. Supported by the Catalan Research Council under grant 1998SGR00119. Partially supported by the Hungarian Research Foundation (OTKA), grant no. T029132.     

Some new row-complete Latin squares

Row-complete Latin squares of orders 9, 15, 21 and 27 are given. The square of order 9 is the smallest possible odd order row-complete Latin square.     

Latin squares over Abelian groups

In this paper, parametric families of Latin squares over Boolean vectors and prime fields constructed earlier are generalized to the case of Abelian groups. Some criteria for realizability of this construction are presented. Some classification results are also given. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 65–71, 2006.     

Latin squares with restricted transversals

We prove that for all odd m ≥ 3 there exists a latin square of order 3 m that contains an ( m ? 1 ) × m latin subrectangle consisting of entries not in any transversal. We prove that for all even n ≥ 10 there exists a latin square of order n in which there is at least one transversal, but all transversals coincide on a single entry. A corollary is a new proof of the existence of a latin square without an orthogonal mate, for all odd orders n ≥ 11 . Finally, we report on an extensive computational study of transversal‐free entries and sets of disjoint transversals in the latin squares of order n ? 9 . In particular, we count the number of species of each order that possess an orthogonal mate. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:124‐141, 2012     

Quarter-regular biembeddings of Latin squares

We apply a recursive construction for biembeddings of Latin squares to produce a new infinite family of biembeddings of cyclic Latin squares of even side having a high degree of symmetry. Reapplication of the construction yields two further classes of biembeddings.     

On a statistical optimality of magic squares

It is shown that an optimal way of running a simple linear regression with n² equally spaced levels in an n × n square is to distribute the levels in the square in the form of a magic square.     

Constructing all magic squares of order three

We find by applying MacMahon's partition analysis that all magic squares of order three, up to rotations and reflections, are of two types, each generated by three basis elements. A combinatorial proof of this fact is given.     

Self-complementary magic squares of doubly even orders

A magic square $M$ in which the entries consist of consecutive integers from $1,2,\dots ,n^2$ is said to be self-complementary of order$n$ if the resulting square obtained from $M$ by replacing each entry $i$ by $n^2+1?i$ is equivalent to $M$ (under rotation or reflection). We present a new construction for self-complementary magic squares of order $n$ for each $n\ge 4$, where $n$ is a multiple of $4$.     

A construction for doubly pandiagonal magic squares

In this note, a doubly magic rectangle is introduced to construct a doubly pandiagonal magic square. A product construction for doubly magic rectangles is also presented. Infinite classes of doubly pandiagonal magic squares are then obtained, and an answer to problem 22 of [G. Abe, Unsolved problems on magic squares, Discrete Math. 127 (1994) 3] is given.     

Intercalates and discrepancy in random Latin squares

An intercalate in a Latin square is a 2 × 2 Latin subsquare. Let be the number of intercalates in a uniformly random n × n Latin square. We prove that asymptotically almost surely , and that (therefore asymptotically almost surely for any ). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in 2 fixed rows of a random Latin square. In addition, we discuss a problem of Linial and Luria on low‐discrepancy Latin squares.     

Generalized Latin squares and their defining sets

A generalized Latin square of type (n,k) is an n×n array of symbols 1,2,…,k such that each of these symbols occurs at most once in each row and each column. Let d(n,k) denote the cardinality of the minimal set S of given entries of an n×n array such that there exists a unique extension of S to a generalized Latin square of type (n,k). In this paper we discuss the properties of d(n,k) for k=2n-1 and k=2n-2. We give an alternate proof of the identity d(n,2n-1)=n²-n, which holds for even n, and we establish the new result . We also show that the latter bound is tight for n divisible by 10.