Complexity of Jelly-No and Hanano games with various constraints

2025-01-21

AI Summary

This paper systematically investigates the computational complexity of the gravity-based puzzle games Jelly-No and Hanano. Addressing parameter variations—including board dimensions, number of colors, and special elements (e.g., black jellies)—we employ reductions, Turing machine simulations, and combinatorial game modeling. Our contributions are threefold: (1) We establish that multicolor Jelly-No is PSPACE-complete; (2) we prove that monochrome Jelly-No becomes PSPACE-complete upon introducing black jellies—resolving a key open question regarding its complexity threshold; and (3) we show that both monochrome Jelly-No and Hanano remain NP-hard even on degenerate 1×n or n×1 boards. Collectively, these results fully characterize the complexity transition of Jelly-No—from NP-hardness to PSPACE-completeness—and unify the hardness boundaries of Jelly-No and Hanano, thereby settling a long-standing open problem in the computational analysis of gravity-based puzzles.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Computational Creativity

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Game Complexity
• Color Variations
• Special Elements

Innovation

• Monochromatic Complexity
• Black Jelly Non-Merging
• Hanano Game Challenge