Igra sparivanja na konačnim podgrafovima pravilnih rešetki

master's thesis

• Author: Marija Pijević
• Mentor: Tomislav Došlić (mentor)
• Committee member: Tomislav Došlić (predsjednik povjerenstva)
• Committee member: Mladen Vuković (član povjerenstva)
• Committee member: Željka Milin-Šipuš (član povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-09-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu pronašli smo pobjedničke strategije za neke oblike heksagonalnih životinja. Na početku smo naveli osnovne pojmove vezane uz teoriju kombinatornih igara i teoriju grafova koji su nam bili potrebni za razradu teme. Nakon toga smo naveli neke oblike heksagonalnih životinja na kojima ćemo raditi sparivanje i objasnili razliku između simetrija \(C_i\) i \(D_i\), \(i = 1, 2, 3, 6\). Posljednje poglavlje posvećeno je analizi nekih simetrija. Najprije smo odigrali igru sparivanja na lančanim poliheksima. Vidjeli smo da pobjednik može biti i prvi i drugi igrač, ovisno o tome je li lančana heksagonalna životinja sastavljena od parnog ili neparnog broja šesterokuta. Nakon toga smo analizirali simetrije \(C_6\) i \(D_6\) i vidjeli da na njima uvijek pobjeđuje samo drugi igrač. Simetrije \(C_2\) i \(D_2\) imaju pobjedničke strategije za oba igrača. Analizu simetrije \(D_1\) smo dosta opsežno napravili i pokazali da pobjednik može biti bilo koji igrač. Simetrija \(C_1\) nije pretjerano zanimljiva pa smo odigrali samo jednu igru i vidjeli da je pobjednik prvi. Grafovi simetrije \(D_3\), na kojima smo igrali igru sparivanja, donose pobjedu drugom igraču. Za simetriju \(C_3\) smo naveli dva grafa, ali nismo igrali igru sparivanja jer je poprilično komplicirano. Nakon svega, došli smo do zaključka. U igri u kojoj se centar simetrije nalazio u središtu središnjeg šesterokuta, pobjednik je uvijek bio drugi igrač. Ako je centar simetrije na polovištu nekog brida, pobjednik je uvijek prvi igrač.

Abstract (english)

In this graduate thesis we found winning strategies for some forms of hexagonal animals. At the beginning, we have provided the basic concepts of combinatorial game theory and theory of graphs which was necessary for analysing the thesis. After that, we pointed out some forms of hexagonal animals on which we will make matching games and we explained the difference between symmetries \(C_i\) and \(D_i, i = 1, 2, 3, 6\). The last chapter was dedicated to analysis of some symmetries. First, we played matching game on the chain polihexes. We have seen that the winner may be the first or the second player, depending on whether the chain hexagonal animal is made up of even or odd number of hexagons. After that, we analyzed symmetries \(C_6\)and \(D_6\) and saw that always wins only the second player. Symmetries \(C_2\) and \(D_2\) have winning strategies for both players. We made analysis of symmetry \(D_1\) quite extensively and showed that the winner can be any player. Symmetry \(C_1\) is not very interesting, so we played only one game and saw that the winner is first player. Graphs of symmetry \(D_3\), on which we played matching game, give victory to the second player. For symmetry \(C_3\) we pointed out two graphs, but we didn’t played matching game because it is quite complicated. After all that, we came to a conclusion. In a game in which center of symmetry is located in the center of the central hexagon, the winner is always the second player. If the center of symmetry is at midpoint of an edge, the winner is always the first player.

• Keywords: igra sparivanja; oblici heksagonalnih životinja; analiza simetrije
• Keywords (english): matching game; forms of hexagonal animals; symmety analysis
• Language: croatian
• URN:NBN: urn:nbn:hr:217:883009
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-01-29 13:34:36