20.500.12556/DKUM-75264 Igra Križci in krožci Tic tac toe game 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. 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. Pozicijska igra Križci in krožci strategija šibka zmaga močan remi Positional game Tic-Tac-Toe strategy weak win strong draw. true true false [M. Fras] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2019-10-07 18:18:07 2019-11-26 13:44:17 2022-08-09 11:07:40 2019-12-10 08:30:49 2019 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 37.091.3:519.1(043.2) 24940296 URN:SI:UM:DK:MU4VU1NG MAG_Fras_Martin_2019.pdf MAG_Fras_Martin_2019.pdf 1 9B804F5B9C4CBB4893EA6E1438823C48 4d3d3662632eabf21699c92e05ef44f6c758a6a58d24932077105c4816a99d7b 13471b2c-7c11-11eb-bb7a-00155d0001ca 20.500.12556/dkum/75c52b2d-17d7-4d2d-ac2d-18f25c067168 https://dk.um.si/Dokument.php?lang=slv&id=138924 Fakulteta za naravoslovje in matematiko 0 0 0