Complexity of Jelly-No and Hanano games with various constraints
Owen Crabtree, Valia Mitsou
TL;DR
The paper resolves the complexity of Jelly-No and Hanano under varying board sizes and colour counts by proving $PSPACE$-Completeness for Jelly-No with an unbounded number of colours, and $PSPACE$-Completeness for the one-colour variant when black jellies are allowed. It also shows that 1-colour Jelly-No and Hanano remain $NP$-Hard under fixed height/width constraints. The main technique combines reductions from the Nondeterministic Constraint Logic (NCL) problem with visibility representations to simulate vertex gadgets and edge flows within Jelly-No, adapting to gravity and blooming mechanics. These results close open questions about the hardness of Jelly-No with multiple colours and clarify how gravity and interaction rules influence the boundary between $NP$ and $PSPACE$ in these puzzles, with implications for the design and analysis of similar gravity-based entanglement games.
Abstract
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 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 coloured blocks of a same colour, which also happens when they make contact. In Hanano the goal is to make all the coloured blocks bloom by making contact with flowers of the same colour. Jelly-No was proven by Chao Yang to be NP-Complete under the restriction that all movable blocks are 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 are the same colour. However, the question whether Jelly-No for more than one colours is also PSPACE-complete or if it too stays in NP 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 do not need to be merged), 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 small constants.