Graeco-Latin squares with embedded balanced superimpositions of Youden squares

Graeco-Latin Squares and a Mistaken Conjecture of Euler∗

Late in his long and productive career, Leonhard Euler published a hundred-page paper detailing the properties of a new mathematical structure: Graeco-Latin squares. In this paper, Euler claimed that a Graeco-Latin square of size n could never exist for any n of the form 4k+2, although he was not able to prove it. In the end, his difficulty was validated. Over a period of 200 years, more than t...

Graeco-Latin Squares and a Mistaken Conjecture of Euler

Latin Squares: Transversals and counting of Latin squares

Author: Jenny Zhang First, let’s preview what mutually orthogonal Latin squares are. Two Latin squares L1 = [aij ] and L2 = [bij ] on symbols {1, 2, ...n}, are said to be orthogonal if every ordered pair of symbols occurs exactly once among the n2 pairs (aij , bij), 1 ≤ i ≤ n, 1 ≤ j ≤ n. Now, let me introduce a related concept which is called transversal. A transversal of a Latin square is a se...

Streamlining Local Search for Spatially Balanced Latin Squares

Streamlined constrained reasoning powerfully boosts the performance of backtrack search methods for finding hard combinatorial objects. We use so-called spatially balanced Latin squares to show how streamlining can also be very effective for local search: Our approach is much faster and generates considerably larger spatially balanced Latin squares than previously reported approaches (up to ord...

Lifting Redundancy from Latin Squares to Pandiagonal Latin Squares

In the pandiagonal Latin Square problem, a square grid of size N needs to be filled with N types of objects, so that each column, row, and wrapped around diagonal (both up and down) contains an object of each type. This problem dates back to at least Euler. In its specification as a constraint satisfaction problem, one uses the all different constraint. The known redundancy result about all dif...