增值遊戲(Drop-or-Hop)是林敬翔(2020)提出的一種在一維棋盤上玩的 不偏混合遊戲。這類遊戲在組合賽局理論(combinatorial game theory)裡有更 詳細的介紹。當雙方都用最有效率的方式進行遊戲時，每個盤面都會有對應的 尼姆值(nim value)，我們可以透過求出它來決定現在的勝負狀態。因此如果 能讓計算尼姆值的過程更為快速，我們便能事先找出更好的勝利策略。
林敬翔(2020)已經給出增值遊戲原始版本的完整分析。在本論文中我們想 出了幾種針對不同方向修改規則的變形，使得原本的分析無法繼續適用。對此 我們同樣去對這些變形進行分析，找到了不同的計算方式，讓我們還是能有效 率地找到尼姆值。

Drop-or-Hop game is an impartial combinatorial game played on a one- dimensional board, first introduced by Lin (2020). This type of games is exten- sively studied in combinatorial game theory. When both players make optimal moves, each board configuration corresponds to a specific nim value, which de- termines whether the current position is winning or losing. Therefore, if we can accelerate the computation of nim values, we can design better winning strategies in advance.
Lin (2020) has already provided a complete analysis of the original version of the game. In this thesis, we introduce several variants of the game with modified rules, rendering the original analysis inapplicable. We analyze these new variants and develop alternative computational methods, allowing us to efficiently determine their nim values.