Complexity of Jelly-No and Hanano games with various constraints
Pith reviewed 2026-05-23 05:16 UTC · model grok-4.3
The pith
Jelly-No is PSPACE-complete with any number of colors, and NP-hard even on constant-width boards.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Jelly-No is PSPACE-complete with an unbounded number of colours. If black jellies are allowed, the game is PSPACE-complete even for one colour. One-colour Jelly-No and Hanano remain NP-hard even if the width or the height of the board are constants.
What carries the argument
Polynomial-time reductions from known PSPACE-hard and NP-hard problems that simulate gravity, block movement, same-color merging, and flower contact under the given color and dimension limits.
If this is right
- Jelly-No instances with multiple colors cannot be solved in polynomial time unless PSPACE equals P.
- One-color Jelly-No becomes PSPACE-complete once black jellies are introduced.
- Fixing board width or height does not drop one-color Jelly-No or Hanano into P.
- The hardness results separate cleanly by color count and by the presence of non-merging blocks.
Where Pith is reading between the lines
- Similar reduction techniques could transfer to other gravity puzzles that use colored blocks and contact rules.
- Level designers may need to further restrict colors or add global limits to keep verification feasible.
- Varying platform count or movement speed in addition to width might reveal new complexity thresholds.
Load-bearing premise
The polynomial-time reductions from known PSPACE-hard or NP-hard problems correctly encode the game rules, gravity, merging, and color constraints of Jelly-No and Hanano under the stated board-size and color restrictions.
What would settle it
Discovery of a polynomial-time algorithm for Jelly-No on boards with multiple colors would falsify the PSPACE-completeness claim.
read the original 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, 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that Jelly-No is PSPACE-complete with an unbounded number of colours, that Jelly-No with black jellies is PSPACE-complete even for one colour, and that one-colour Jelly-No and Hanano remain NP-hard when the board width or height is held constant. These results build on prior NP-completeness and PSPACE-completeness results for the same games under single-colour restrictions and close the open question of PSPACE-completeness for multi-colour Jelly-No.
Significance. The results provide a more complete complexity landscape for Jelly-No and Hanano by establishing PSPACE-completeness in the multi-colour and one-colour-with-black-jellies settings and by showing that NP-hardness persists under constant-dimension restrictions. The proofs rely on polynomial-time reductions from known PSPACE-hard and NP-hard problems that correctly encode the games' gravity, merging, and colour-interaction rules.
minor comments (2)
- The abstract and introduction could more explicitly state the precise board-size and colour restrictions under which each theorem holds (e.g., §1, paragraph 3).
- Figure captions for the reduction gadgets should include a brief reminder of which game rules (gravity, merging, black jellies) each gadget encodes.
Simulated Author's Rebuttal
We thank the referee for their positive assessment and recommendation to accept the manuscript. The report contains no major comments requiring response.
Circularity Check
No significant circularity; reductions from independent prior results
full rationale
The paper's central claims are PSPACE-completeness and NP-hardness results obtained via explicit polynomial-time reductions from known hard problems (e.g., from prior work by Chao Yang and Michael C. Chavrimootoo on the same games under different constraints). These source problems are established by different authors and are independent of the present reductions. No self-citation is load-bearing, no parameter is fitted and then renamed as a prediction, no ansatz is smuggled, and no uniqueness theorem is invoked from the authors' own prior work. The derivation chain consists of gadget constructions that encode gravity, merging, and color rules; these steps are externally falsifiable and do not reduce to the target claim by definition. The manuscript is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Polynomial-time reductions preserve PSPACE-hardness and NP-hardness
discussion (0)
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