Swish is a card game in which players are given cards having symbols (hoops and balls), and find a valid superposition of cards, called a "swish." Dailly, Lafourcade, and Marcadet (FUN 2024) studied a generalized version of Swish and showed that the problem is solvable in polynomial time with one symbol per card, while it is NP-complete with three or more symbols per card. In this paper, we resolve the previously open case of two symbols per card, which corresponds to the original game. We show that Swish is NP-complete for this case. Specifically, we prove the NP-hardness when the allowed transformations of cards are restricted to a single (horizontal or vertical) flip or 180-degree rotation, and extend the results to the original setting allowing all three transformations. In contrast, when neither transformation is allowed, we present a polynomial-time algorithm. Combining known and our results, we establish a complete characterization of the computational complexity of Swish with respect to both the number of symbols per card and the allowed transformations.

翻译：Swish是一种卡牌游戏，玩家获得带有符号（圆环和圆球）的卡片，并需要找到一种有效的卡片叠加方式，称为“swish”。Dailly、Lafourcade和Marcadet（FUN 2024）研究了Swish的广义版本，证明当每张卡片仅有一个符号时该问题可在多项式时间内求解，而当每张卡片有三个或更多符号时该问题是NP完全的。本文解决了先前悬而未决的每张卡片两个符号的情况，这对应原始游戏规则。我们证明Swish在此情况下同样是NP完全的。具体而言，我们证明了当卡片允许的变换仅限于单次（水平或垂直）翻转或180度旋转时的NP困难性，并将结果推广到允许所有三种变换的原始设定。与之相对，当不允许任何变换时，我们提出了一种多项式时间算法。结合已知结果与我们的发现，我们建立了Swish计算复杂性关于每张卡片符号数量与允许变换方式的完整刻画。