On three-rowed Chomp

Chomp is a 50 year-old game played on a partially ordered set P. It has been in the center of interest of several mathematicians since then. Even when P is simply a 3 × n lattice, we have almost no information about the winning strategy. In this paper we present a new approach and a cubic algorithm for computing the winning positions for this case. We also prove that from the initial positions there are infinitely many winning moves in the third row

On three-rowed Chomp

Chomp is a 50 year-old game played on a partially ordered set P. It has been in the center of interest of several mathematicians since then. Even when P is simply a 3 × n lattice, we have almost no information about the winning strategy. In this paper we present a new approach and a cubic algorithm for computing the winning positions for this case. We also prove that from the initial positions there are infinitely many winning moves in the third row

Multi-Dimensional Chocolate and Nim with a Pass

Chocolate-bar games are variants of the CHOMP game. A three-dimensional chocolate bar comprises a set of cubic boxes sized 1 X 1 X 1, with a bitter cubic box at the bottom of the column at position (0,0). For non-negative integers u,w such that u < x and w \< z, the height of the column at position (u,w) is min (F(u,w),y) +1, where F is a monotonically increasing function. We denote this chocolate bar as CB(F,x,y,z). Each player, in turn, cuts the bar on a plane that is horizontal or vertical along the grooves, and eats the broken piece. The player who manages to leave the opponent with the single bitter cubic box is the winner. In this study, functions F such that the Sprague--Grundy value of CB(F,x,y,z) is x xor y xor z are characterized. We have already published the research on three-dimensional chocolate games. In this paper, the authors study a multi-dimensional chocolate game, where the dimension is more than three, and apply the theory to the problem of pass move in Nim. We modify the standard rules of the game to allow a one-time pass, that is, a pass move that may be used at most once in the game and not from a terminal position. Once a pass has been used by either player, it is no longer available. It is well-known that in classical Nim, the introduction of the pass alters the underlying structure of the game, significantly increasing its complexity. A multi-dimensional chocolate game can show a perspective on the complexity of the game of Nim with a pass. Therefore, the authors address a longstanding open question in combinatorial game theory. The authors present this paper, since their discovery seems to be significant. It appears to the authors that the relation between chocolate games and Nim with a pass will be an essential research topic soon.Comment: 17 page

Chomp-pelistä

Tämä tutkielma käsittelee kombinatorista peliä Chomp, jonka pelialue voidaan ajatella suklaalevynä, josta alin vasemmanpuoleisin pala on myrkyllinen. Pelaajat poistavat vuorotellen osioita suklaalevystä ja pelaaja, joka joutuu poistamaan myrkyllisen palan, häviää. Tutkielmassa käsitellään sekä kombinatoristen pelien teoriaa että erilaisten Chomp-pelien tuloksia. Tutkielman alkuosa keskittyy kombinatoristen pelien teoriaan. Aluksi määritellään äärellinen ja ääretön kombinatorinen peli ja esitetään näiden ominaisuuksia. Tämän jälkeen osoitetaan, miten äärellinen peli voidaan laajentaa vastaavaksi äärettömäksi peliksi. Lisäksi esitetään pelin alkaminen keskeneräisestä pelitilanteesta. Kombinatorisiin peleihin liittyen määritellään strategian ja voittostrategian käsitteet sekä käydään läpi näihin liittyviä tuloksia. Esitetään myös strategian soveltaminen keskeneräiseen pelitilanteeseen. Alkuosan loppupuolella todistetaan yksi kombinatoristen pelien tärkeä lause, Galen ja Stewartin lause, jonka mukaan jokaisessa äärettömässä kombinatorisessa pelissä jommalla kummalla pelaajalla on voittostrategia. Tutkielman loppuosa keskittyy kombinatoriseen peliin Chomp ja sen ominaisuuksiin. Aluksi määritellään pelin Chomp perusversio ja osoitetaan miten se voidaan tulkita aiemmin määritellyksi kombinatoriseksi peliksi. Seuraavaksi osoitetaan, että Chomp-pelissä aloittavalla pelaajalla on aina voittostrategia, jos pelialue sisältää suurimman alkion. Tämän jälkeen esitetään esimerkkejä tavanomaisista kaksiulotteisista Chomp-peleistä ja niiden tilanteista. Näistä kolmerivisen Chomp-pelin eri tilanteiden voittostrategiat käsitellään hieman tarkemmin. Perusversion käsittelyn jälkeen laajennetaan Chomp moniulotteiseksi, ja esitetään esimerkkejä kolmiulotteisesta ja neliulotteisesta pelialueesta. Lopuksi laajennetaan sekä kaksiulotteinen että kolmiulotteinen versio myös äärettömäksi ja tutkitaan voittostrategioita näissä pelialueissa

Combinatorial Games with a Pass: A dynamical systems approach

By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game's underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations we are able to identify underlying structural connections between these "games with passes" and a recently introduced class of "generic (perturbed) games." This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game.Comment: 39 pages, 13 figures, published versio

Dvije igre i njihova generalizacija

U članku prezentiramo igre Nim i Chomp i njihovu generalizaciju. Prva igra poznata je čitateljima math.e iz članka Matka Botinčana Kombinatorne igre, objavljenog u šestom broju, i ima jednostavnu pobjedničku strategiju. Za drugu igru može se dokazati da igrač koji je prvi na potezu ima pobjedničku strategiju, ali je njezin opis poznat samo u nekim specijalnim slučajevima. Na obje igre odnosi se teorem S. Byrnesa o periodičnosti igara na parcijalno uređenim skupovima koji je autoru, tada srednjoškolcu, priskrbio stipendiju od 100000 američkih dolara

Automated conjecturing II : chomp and reasoned game play

We demonstrate the use of a program that generates conjectures about positions of the combinatorial game Chomp—explanations of why certain moves are bad. These could be used in the design of a Chomp-playing program that gives reasons for its moves. We prove one of these Chomp conjectures—demonstrating that our conjecturing program can produce genuine Chomp knowledge. The conjectures are generated by a general purpose conjecturing program that was previously and successfully used to generate mathematical conjectures. Our program is initialized with Chomp invariants and example game boards—the conjectures take the form of invariant-relation statements interpreted to be true for all board positions of a certain kind. The conjectures describe a theory of Chomp positions. The program uses limited, natural input and suggests how theories generated on-the-fly might be used in a variety of situations where decisions—based on reasons—are required