Eulerjev problem 36 častnikov
delo diplomskega seminarja
Abstract
Latinski kvadrat reda $n$ je tabela velikosti $n \times n$, sestavljena iz elementov množice moči $n$, v kateri je vsak od elementov zastopan v vsaki vrstici in v vsakem stolpcu. Dva latinska kvadrata reda $n$ sta ortogonalna, če njuna superpozicija tvori same različne urejene pare. Nastalemu kvadratu rečemo grško-latinski kvadrat reda $n$. Eulerjev problem $36$ častnikov, ki se sprašuje, ali je možno razporediti $36$ častnikov iz šestih različnih regimentov in šestih različnih činov, v formacijo $6 \times 6$, tako da je v vsaki vrsti in vsaki koloni zastopan vsak regiment in vsak čin, je potem enak vprašanju obstoja grško-latinskega kvadrata reda šest. Tega lahko prevedemo v vprašanje obstoja transverzalnega načrta $TD(4, 6)$, za katerega lažje dokažemo, da ne obstaja. Grško-latinske kvadrate lihih redov znamo enostavno konstruirati, prav tako poznamo kvadrata reda štiri in osem. Dejstvo, da iz dveh grško-latinskih kvadratov redov $n_1$ in $n_2$ dobimo grško-latinski kvadrat reda $n_1 \times n_2$, pa nam pomaga konstruirati še kvadrate višjih redov oblike $n \not\equiv 2\pmod{4}$. Euler je domneval, da grško-latinski kvadrati preostalih redov ne obstajajo, vendar je bila njegova domneva ovržena skoraj dvesto let kasneje. Dva načina konstrukcije takih kvadratov sta s pomočjo ortogonalnih tabel in Wilsonove konstrukcije.
Keywords
matematika;ortogonalni latinski kvadrati;grško-latinski kvadrati;ortogonalne tabele;transverzalni načrti;
Data
| Language: | Slovenian |
|---|---|
| Year of publishing: | 2023 |
| Typology: | 2.11 - Undergraduate Thesis |
| Organization: | UL FMF - Faculty of Mathematics and Physics |
| Publisher: | [K. Kranjec] |
| UDC: | 519.1 |
| COBISS: |
165831171
|
| Views: | 735 |
| Downloads: | 37 |
| Average score: | 0 (0 votes) |
| Metadata: |
|
Other data
| Secondary language: | English |
|---|---|
| Secondary title: | Euler’s 36 Officers Problem |
| Secondary abstract: | A Latin square of order $n$ is an $n \times n$ array of elements from a set of size $n$ in which each element occurs in every row and every column. Two Latin squares of order $n$ are orthogonal if their superposition yields unique ordered pairs. The resulting square is then called a Graeco-Latin square of order $n$. Euler’s $36$ Officers Problem which poses a question if it is possible to arrange $36$ officers of six different regiments and of six different ranks in a formation $6 \times 6$ where each row and each file contains one officer of each regiment and one of each rank, is equal to the question of existence of Graeco-Latin square of order six. In design theory this question translates to the question of existence of a transversal design $TD(4, 6)$ the non-existence of which is easier to prove. Graeco-Latin squares of odd orders are easy to construct as well as squares of orders four and eight. The fact that a Graeco-Latin square of order $n_1 \times n_2$ can be constructed from two Graeco-Latin squares of orders $n_1$ and $n_2$ helps us construct squares of higher orders $n$ where $n \not\equiv 2\pmod{4}$. Euler conjectured that there exist no Graeco-Latin squares of other orders which was disproven almost two hundred years later. Two ways of constructing such squares are using orthogonal tables and Wilson’s construction. |
| Secondary keywords: | orthogonal latin squares;Graeco-Latin squares;orthogonal arrays;transversal designs; |
| Type (COBISS): | Final seminar paper |
| Study programme: | 0 |
| Embargo end date (OpenAIRE): | 1970-01-01 |
| Thesis comment: | Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja |
| Pages: | 30 str. |
| ID: | 19981804 |
Recommended works: