Kuadrat Graeco-Latin

Kuadrat Graeco-Latin n×n nyaéta tabel nu unggal sel-na mibanda pasangan simbol, susunan simbol tina unggal dua set unsur n. éach pair occurs exactly once in the table. éach symbol in the two, not necessarily distinct, sets occurs exactly once in éach row and exactly once in éach column. Kuadrat Graeco-Latin dipaké dina desain percobaan.

A 4×4 Graeco-Latin square on the sets {A, B, C, D} and {α, β, γ, δ} is:

A αB γC δD βB βA δD γC αC γD αA βB δD δC βB αA γ

The tabular arrangements of {A, B, C, D} (Latin characters) alone and {α, β, γ, δ} (Greek characters) alone éach forms a Latin square. éach pair from the two sets (i.e. every element of their cartesian product) occurs exactly once and we say that the two Latin squares are orthogonal.

History

In the 1780s, Leonard Euler  demonstrated methods for constructing Graeco-Latin squares where n is odd or a multiple of 4. He further proved that no 2×2 square exists and conjectured that none existed for n=4k+2, where k is a natural number.

In 1901, Gaston Tarry demonstrated that there was no 6×6 square by enumerating all the possible arrangements of symbols. However, in 1959, Parker, Bose and Shrikhande constructed a 10×10 square.

In 1978, the French writer Georges Perec used the 10×10 square (believed then to be the only one possible) for the structure of constraints underlying his novel Life: A User's Manual.

Tempo ogé
 Latin square

Kategori:Matematika