Distance in Latin Squares

On the Number of Maximum Inner Distance Latin Squares

“…Since Row A contains only 0's and 1's, all of its neighbors have no consecutive -1's by Lemma 3.8. The only remaining rows that satisfy this are type 2's with h 1], and both of these values work as one may check. For h ′ = 3 − n 2 , m ′ ranges in [0, 2] but only m ′ = 1 works as the others contain 2 consecutive -1's.…”

“…The symbols (i, j) will denote the i-th row and j-th column in a matrix. Cells are adjacent if they share an edge vertically or horizontally, for example cells (1,3) and (2,3) share an edge vertically, but cells (1,2) and (2,3) Figure 2: Two Latin squares of order 6, of inner distance 2 (left) and 1 (right).…”

“…In this paper we also establish existence for all smaller inner distances. Much of our introductory work is showcased in [1], with a primary focus on determining the maximum inner distance for Latin squares and Sudoku Latin squares. In this paper, we focus on enumerating maximum inner distance squares, and calculated the exact number for any n:…”

“…Where P (n) = 1 4 n 2 + 2 if n ≡ 0 (mod 4), and P (n) = 1 4 n 2 + 1 if n ≡ 2 (mod 4). The odd case is easy, shown in [1].…”

“…Where P (n) = 1 4 n 2 + 2 if n ≡ 0 (mod 4), and P (n) = 1 4 n 2 + 1 if n ≡ 2 (mod 4). The odd case is easy, shown in [1]. The result for evens is computed in this paper by a connection to graph theory, and direct computation.…”