Igra Križci in krožci

Fras, Martin

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Title:	Igra Križci in krožci
Authors:	ID Fras, Martin (Author) ID Brešar, Boštjan (Mentor) More about this mentor...
Files:	MAG_Fras_Martin_2019.pdf (1,24 MB) MD5: 9B804F5B9C4CBB4893EA6E1438823C48 PID: 20.500.12556/dkum/75c52b2d-17d7-4d2d-ac2d-18f25c067168
Language:	Slovenian
Work type:	Master's thesis/paper
Typology:	2.09 - Master's Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	V magistrskem delu predstavljamo osnovne pojme kombinatoričnih iger na treh variacijah igre Križci in krožci: simetrične igre, igre tipa Izdelovalec-Lomilec in igre tipa berač. Obravnavamo igre Križci in krožci zrazličnimi dimenzijami igralnega polja in predstavimo različne strategije igralcev ter opazujemo odnose med spremembo dimenzije igralnega polja ter rezultatom igre. Ugotavimo, da v simetrični igri na polju velikosti $n^d$ z uporabo ustrezne strategije zmaga Prvi igralec, če je z uporabo iste strategije zmagal na polju $n^k,\ k<d$. V igrah tipa Izdelovalec-Lomilec ugotovimo, da večanje dimenzije igralnega polja ne more škodovati Izdelovalcu, škodi pa lahko Lomilcu, kar je posledica večanja števil zmagovalnih vrst, v katerih je vsebovana posamezna celica. V igrah tipa berač ugotovimo, da lahko Prvi igralec, na polju velikosti $(2n-1)^d,\ n\geq 4,\ d\geq 2$, s pomočjo zrcaljenja preko sredinske celice igralnega polja doseže najmanj remi. Z zrcaljenjem preko središča igralnega polja lahko vsaj remi doseže tudi Drugi igralec na poljih dimenzij $2n^d,\ n\geq 1,\ d\geq 2$. Na koncu predstavimo še igre Neomejen $n$ v vrsto, Hex, Bridge-it in Izdelovalec-Lomilec dominacijske igre, ki si delijo nekaj pomembnih lastnosti z igro Križci in krožci.
Keywords:	Pozicijska igra, Križci in krožci, strategija, šibka zmaga, močan remi
Place of publishing:	Maribor
Publisher:	[M. Fras]
Year of publishing:	2019
PID:	20.500.12556/DKUM-75264
UDC:	37.091.3:519.1(043.2)
COBISS.SI-ID:	24940296
NUK URN:	URN:SI:UM:DK:MU4VU1NG
Publication date in DKUM:	26.11.2019
Views:	1674
Downloads:	178
Metadata:
Categories:	FNM
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Licences

License:	CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

Link:	http://creativecommons.org/licenses/by-nc-nd/4.0/
Description:	The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.
Licensing start date:	07.10.2019

Secondary language

Language:	English
Title:	Tic tac toe game
Abstract:	In this thesis, we present basic terms of combinatorial games on three variants of Tic-Tac-Toe game: symmetric game, Maker-Breaker game and Misere. We use these variants on different dimensions of the playing field to present different player strategies, while observing changes of the game outcome depending on the change of the game's playing field. We concluded, that the symmetric game on the playing field size of $n^d$ is a First player strategy win, if First player won by using the same strategy on a playing field size of $n^k,\ k<d$. In Maker-Breaker games, increasing the dimension of the playing field can not harm Maker, but it can harm Breaker, which is the result of higher number of winning sets in higher dimensions. In Misere games, we establish that First player, if using mirroring strategy, can achieve at least a draw on the playing field size of $(2n-1)^d,\ n\geq 4,\ d\geq 2$, while win is achieved if the playing field has no final drawing position. Similarly, it is possible for Second player to achieve at least a draw, using similar mirroring strategy, if the playing field is of size $2n^d,\ n\geq1,\ d\geq 2$. At the end of the thesis we additionally present games of Unlimited $n$-in-a-row, Hex, Bridge-it and Maker-Breaker domination games, which are similar to the Tic-Tac-Toe games.
Keywords:	Positional game, Tic-Tac-Toe, strategy, weak win, strong draw.

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