AI 中文总结
本文研究巧克力游戏这一尼姆游戏的推广，证明“美味条件”可作为多种形状巧克力棒的斯普莱格-格伦迪值由各维度尺寸尼姆和给出的条件。
AI 中文摘要
本文研究巧克力游戏。在巧克力游戏中，每位玩家将一块巧克力棒切成两块，吃掉其中一块，且不能吃掉指定的苦味块；吃掉苦味部分的玩家输掉游戏。该游戏可视为尼姆（Nim）游戏的推广，已有研究探讨了对于何种形状的巧克力棒，其局面的斯普莱格-格伦迪（Sprague--Grundy）值可通过巧克力棒宽度与高度的尼姆和（异或，XOR）计算，更高维的情况也已被研究。本文证明，前人针对递增阶梯形巧克力棒引入的“美味条件”，也可作为多种其他形状巧克力棒的充分条件，在某些情况下甚至是充要条件，使得斯普莱格-格伦迪值可由巧克力棒各维度尺寸的尼姆和给出。
英文摘要
In this paper, we investigate Chocolate Games. In Chocolate Games, each player cuts a chocolate bar into two chocolate bars and eats one of them so that they do not eat an indicated bitter block. The player who eats the bitter part loses the game. This game can be considered as a generalization of Nim, and previous studies consider the condition that for what kind of the shape of chocolate bar, the Sprague--Grundy value of the position can be calculated by Nim--sum (XOR) of the width and height of the chocolate bar. Higher dimensional cases were also studied. In this paper, we show that the ``Tasty condition,'' which was introduced in previous work for increasing staircase chocolate bars, can also be used for various other shapes of chocolate bars as a sufficient condition, or in some cases a necessary and sufficient condition, for the Sprague--Grundy value to be given by the Nim-sum of the sizes of the chocolate bar in each dimension.
Comments30 pages, 21 figures
