Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture

Abstract

Ifis the prime power decomposition of an integer v, and we define the arithmetic function n(v) bythen it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated asMacNeish conjectured that the actual value of N(v) is n(v).

Mutually Orthogonal Families of Linear Sudoku Solutions

“…Beginning in 1782 with Euler's thirty-six officers [6], problems and design applications of orthogonal latin squares have been extremely well-documented, appearing in literature ranging from applied combinatorics books ( [17], for example) to research papers on the construction and size of orthogonal families (see [5] for a good survey). A well-known open problem in this field is the determination of N (n), the maximum size of a family of mutually orthogonal latin squares of order n. It is relatively easy to show that N (n) = n − 1 if n is a power of a prime, but the problem is far more difficult for other values of n. For instance, as an outgrowth of the thirty-six officers problem, Euler conjectured that N (n) = 1 whenever n ≡ 2 mod 4; this stood until 1960 when Bose, Shrikhande, and Parker ( [2] and [3] collectively) showed that Euler's conjecture is false for all n ≡ 2 mod 4, n > 6. Many current results involve providing lower bounds for N (n) (again, see [5]), and Mullen [15] has suggested that the determination of N (n) be regarded as the next 'Fermat' problem.…”

“…showing that any collection of linear Keedwell solutions lies in the same orbit of the sudoku group (Proposition 3.6); [3] Orthogonal sudoku solutions 411…”

Эйлер И Комбинаторика

“…Эту гипотезу Эйлера частично подтвердил в 1900 г. G. Tarry [16], доказав (фактически -перебо-ром всех возможных неэквивалентных пар латинских квадратов размера 6 × 6), что в случае n = 6 она верна. Следующее про-движение в исследовании гипотезы Эйлера было сделано лишь в 1959 г., когда Bose, Shrikhande и Parker [2], [14], [1] установили, что пары ортогональных латинских квадратов существуют при всех натуральных n ̸ = 2, 6.…”

On the theory and application of sum composition of Latin squares and orthogonal Latin squares

“…Although this method does not work for all pairs of Latin squares, it has an immediate application in the construction of orthogonal Latin squares with certain interesting and useful combinatorial structures, including those of order At + 2, t > 2. As will be seen this method is easy, and is simpler than other known methods for the construction of orthogonal Latin squares of order At + 2 (see [1]). Perhaps the idea of sum composition can be extended to other combinatorial structures and designs.…”