This work shows new results on the complexity of games Jelly-No and Hanano with various constraints on the size of the board and number of colours. Hanano and Jelly-No are one-player, 2D side-view puzzle games with a dynamic board consisting of coloured, movable blocks disposed on platforms. These blocks can be moved by the player and are subject to gravity. Both games, created by Qrostar and available online, somehow vary in their gameplay, but the goal is always to move the coloured blocks in order to reach a specific configuration and make them interact with each other or with other elements of the game. In Jelly-No the goal is to merge all blocks of the same colour, which happens when they make contact. In Hanano the goal is to make all the coloured blocks bloom by making contact with flowers that have the same colour. Jelly-No was proven by Chao Yang to be NP-complete under the restriction that all movable blocks have the same colour and NP-hard for more colours. Hanano was proven by Michael C. Chavrimootoo to be PSPACE-complete under the restriction that all movable blocks have the same colour. However, the question of PSPACE-completeness for Jelly-No with more than one colours was left open. In this paper, we settle this question, proving that Jelly-No is PSPACE-complete with an unbounded number of colours. We further show that, if we allow black jellies (that is, jellies that cannot and do not need to merge), the game is PSPACE-complete even for one colour. We further show that one-colour Jelly-No and Hanano remain NP-hard even if the width or the height of the board are constants.

翻译：本文展示了在不同棋盘大小和颜色数量约束下，Jelly-No和Hanano游戏复杂性的新结果。Hanano和Jelly-No是单人、二维侧视解谜游戏，其动态棋盘由分布在平台上的彩色可移动方块组成。玩家可以移动这些方块，并且它们受重力影响。这两款由Qrostar创建并可在线获取的游戏，在玩法上有所差异，但目标都是移动彩色方块以达成特定配置，并使它们彼此互动或与游戏中的其他元素互动。在Jelly-No中，目标是合并所有相同颜色的方块，这发生在它们接触时。在Hanano中，目标是让所有彩色方块通过接触同色的花朵而“绽放”。Chao Yang已证明，在限制所有可移动方块颜色相同的情况下，Jelly-No是NP完全的；对于更多颜色，它是NP难的。Michael C. Chavrimootoo已证明，在限制所有可移动方块颜色相同的情况下，Hanano是PSPACE完全的。然而，关于超过一种颜色时Jelly-No的PSPACE完全性问题一直悬而未决。本文解决了这一问题，证明Jelly-No在颜色数量无界的情况下是PSPACE完全的。我们进一步表明，如果允许黑色果冻（即无法也不需要合并的果冻），该游戏即使在单色情况下也是PSPACE完全的。我们还证明，即使棋盘宽度或高度为常数，单色Jelly-No和Hanano仍然是NP难的。