增殖遊戲 (Drop-or-Hop) 是由林敬翔 (2020) 提出的一
種在有向圖(DAG)上進行的雙人遊戲。此遊戲屬於組合賽
局理論(combinatorial game theory)中所定義的不偏賽局
(impartial game)。在正常遊玩條件下(每位玩家需要在自己
回合內進行一次動作，否則該名玩家輸)，不偏賽局中每
個盤面皆有一個相對應的尼姆值(nim value)，我們可以依
照尼姆值判別當前盤面的勝負。因此，如果可以有效計算
某不偏賽局任意盤面的尼姆值，我們就可以即時的規劃此
遊戲的必勝策略。

林敬翔 (2020) 已給出增殖遊戲在有根有向二分圖上
進行時其尼姆值的完整分析。在此論文中，我們透過改變
增殖遊戲的規則，或使其在不規則的棋盤中進行，找到兩
個相關的遊戲變型。這些變型在其相關有向圖上具有特殊
的性質使其無法囊括在已知的分析之中。對此，我們提出
分別的對應方法，讓我們亦能夠有效地計算出這些變型中
任意盤面的尼姆值。

Drop-or-Hop, proposed by Lin (2020), is a two-player pebble
game played on a directed acyclic graph (DAG). It belongs to the class
of impartial games in combinatorial game theory. Under the normal
play condition (where a player loses if she does not have any available
move at her turn), each configuration of an impartial game would have
an associated nim value, which can be used to determine whether the
configuration is winning or losing. Therefore, if we can effectively com-
pute the nim value of any configuration, we can formulate a winning
strategy of the impartial game in time.

Lin (2020) gave a complete analysis of Drop-or-Hop for the case
when the DAG is a rooted bipartite graph. In this thesis, we consider
two variants of the game, which are obtained by modifying the legal
movement from the previous studies, or introducing minor “irregulari-
ties” to the shape of the board. Since the DAGs for these variants are
no longer rooted bipartite, the previous analysis does not apply. Yet,
we provide alternative analyses, and show that the nim value of any
configuration in these variants can still be determined efficiently.

1 Introduction 1
1.1 Nim Value 2
1.2 Drop-or-Hop Game 3
1.3 3-Color-Drop-or-Hop Game 4
1.4 Drop-or-Hop on Irregular Boards 5
1.5 Thesis Organization 6
2 Nim Values in 3-Color-Drop-or-Hop Game 7
3 Nim Values in Drop-or-Hop Game on Twisted Boards 12
3.1 V-type Graphs 13
3.1.1 When v2 is Black 15
3.1.2 When v2 is White 22
4 Conclusion and Future Work 31

[1] Thomas S. Ferguson. Game Theory, 2nd Edition, 2014.
[2] Charles L. Bouton. “Nim, A Game with a Complete Mathematical Theory.” Annals of Mathematics, 3(14):35–39, 1901–1902.
[3] Martin Gardner. “Mathematical Games: Cram, Crosscram and Quadraphage: New Games having Elusive Winning Strategies.” Scientific American, 230(2):106–108, 1974.
[4] Ching-Hsiang Lin. A Study on Two Self-Invented Pebble Games. Master thesis, National Tsing Hua University, June 2020.