Nepristranski šah na Youngovih diagramih / Impartial Chess on Young Diagrams

V okviru predlaganega projekta načrtujemo preučevanje nepristranske igre, ki povezuje igro šaha in Youngove diagrame.

Nepristranske igre se lahko natančno opiše samo s tem da povemo katere skupne poteze so možne. Namreč, igrata dva igralca, kjer so vse poteze skupne igralcema, in zato ni pomembno kdo začne. Izmenjujejeta se v premikanju, dokler ni več možnih potez. Po konvenciji zadnja poteza prinese zmago.

Youngov diagram, je grafična predstava particije števil, le ta je padajoče zaporedje naravnih števil. Diagram zgleda kot šahovnica odgrizena na večih mestih iz spodnjega desnega kota.

Glavna linija raziskav je povezana z nadaljevanjem obstoječih raziskav, objavljenih v [Gottlieb2023] in [Gottlieb2024]. Tam je bila preučena nepristranska igra, kjer je okrnjen kralj, ki se lahko premika samo navzdol ali desno, dokler se ne zagozdi v kotu Yougovega diagrama. Opazili smo, da je ta igra podobna nepristranskemu šahu, ki ga je predstavil Berlekamp v svojih spletnih videih [BerlekampPage]. Nepristranski šah je nepristranska igra, ki se igra na pravokotnem razdelku s posamezno šahovsko figuro, ki začne na najbolj zgornjem levem položaju. Premik iz položaja (i, j) v položaj (i’, j’) je dovoljen, če (a) spoštuje pravila gibanja določene šahovskega figure, in (b) 0<=i'<= i’, kot tudi 0<=j'<=j. Ideja skupne igre, ki ju generalizira, je

nastala: šahovska figura, ki se premika po Youngovem diagramu.

EN

In the framework of the proposed project, we plan to investigate an Impartial game connecting the game of Chess and Young diagrams.

Impartial games can be precisely described by stating which joint moves are possible. Indeed, two players play, where all moves are shared between them, and thus, it does not matter who starts. They alternate in moving until no more moves are possible. According to convention, the last move brings victory.

A Young diagram is a graphical representation of an integer partition, which is a decreasing sequence of natural numbers. The diagram resembles a chessboard with several pieces removed from the bottom right corner.

The main line of research is related to the continuation of existing research published in [Gottlieb2023] and [Gottlieb2024]. There, an impartial game where a handicapped King which can only move down or right, until stuck in a corner of a Young diagram, was studied. We noticed that this game is similar to Impartial Chess introduced by Berlekamp in his online videos [BerlekampPage]. Impartial Chess is an impartial game played on a rectangular partition with a single chess-piece starting on the top-left most position. The move from a position (i,j) to a position (i’,j’) is allowed if (a) it respects the movement rules of the particular game piece of chess, and (b) 0<=i'<=i’, as well as $0<=j'<=j$. An idea of a joint game that generalized both came to existence: a chess piece that moves along a Young diagram.