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Carrying is Hard: Exploring the Gap between Hardness for NP and PSPACE for the Hanano and Jelly no Puzzles

Michael C. Chavrimootoo, Jin Seok Youn

TL;DR

The paper investigates the boundary between $NP$ and $PSPACE$ for the Hanano and Jelly puzzles under irreversible gravity, showing that Hanano remains $PSPACE$-complete even under restrictive conditions that make Jelly $NP$-complete in some cases. It leverages visibility representations and Nondeterministic Constraint Logic (NCL) to design gadgets (red bend, OR, AND) that simulate NCL edge flips within Hanano, establishing a $PSPACE$-hardness reduction. A central claim is that the ability of blocks to carry each other under gravity is the key mechanism driving the hardness jump, supported by an NP-membership result for width-1 blocks and a polynomial-bounded analysis of configurations via a fall metric $f(C)$ under certain restrictions. The work clarifies when carrying yields intractability and outlines directions for a broader framework to study irreversible gravity puzzles, with contrasts to Jelly and several open questions on further restrictions and complexity classifications.

Abstract

The Hanano Puzzle is a one-player game with irreversible gravity, where the goal is to make colored blocks make contact with flowers of the corresponding color. The game Jelly no Puzzle shares similar mechanics. In general, determining if a given level of each of the two games is solvable is PSPACE-complete. There are also known restrictions under which determining if a level of Jelly no Puzzle is solvable is NP-complete. We find that under the same restrictions, determining if a level of Hanano Puzzle is solvable remains PSPACE-complete. We thus study several restrictions on Hanano, contrast them with known results about Jelly no Puzzle, and posit that the mechanism at the heart of the PSPACE-hardness is the ability for blocks to carry each other.

Carrying is Hard: Exploring the Gap between Hardness for NP and PSPACE for the Hanano and Jelly no Puzzles

TL;DR

The paper investigates the boundary between and for the Hanano and Jelly puzzles under irreversible gravity, showing that Hanano remains -complete even under restrictive conditions that make Jelly -complete in some cases. It leverages visibility representations and Nondeterministic Constraint Logic (NCL) to design gadgets (red bend, OR, AND) that simulate NCL edge flips within Hanano, establishing a -hardness reduction. A central claim is that the ability of blocks to carry each other under gravity is the key mechanism driving the hardness jump, supported by an NP-membership result for width-1 blocks and a polynomial-bounded analysis of configurations via a fall metric under certain restrictions. The work clarifies when carrying yields intractability and outlines directions for a broader framework to study irreversible gravity puzzles, with contrasts to Jelly and several open questions on further restrictions and complexity classifications.

Abstract

The Hanano Puzzle is a one-player game with irreversible gravity, where the goal is to make colored blocks make contact with flowers of the corresponding color. The game Jelly no Puzzle shares similar mechanics. In general, determining if a given level of each of the two games is solvable is PSPACE-complete. There are also known restrictions under which determining if a level of Jelly no Puzzle is solvable is NP-complete. We find that under the same restrictions, determining if a level of Hanano Puzzle is solvable remains PSPACE-complete. We thus study several restrictions on Hanano, contrast them with known results about Jelly no Puzzle, and posit that the mechanism at the heart of the PSPACE-hardness is the ability for blocks to carry each other.
Paper Structure (6 sections, 8 theorems, 2 equations, 5 figures, 1 table)

This paper contains 6 sections, 8 theorems, 2 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Organization.
  3. Preliminaries
  4. One-Color PSPACE-Completeness
  5. Carrying is Hard
  6. Conclusion and Future Work

Key Result

theorem 1

$\rm \textsc{Hanano}$ remains $\PSPACE$-complete even when restricted to no movable gray blocks and all colored blocks/flowers have the same color.

Figures (5)

  • Figure 1: Level 1 of Hanano.
  • Figure 2: Two configurations of Level 1 of Jelly no Puzzle.
  • Figure 3: Example of an NCL graph and its visibility representation, reproduced from cha:j:hanano.
  • Figure 4: Our three gadgets: a red-bend gadget, an OR gadget, and an AND gadget.
  • Figure 5: Adding the width-2 block G2 forces G1 to move right and then return to its original position with no bloom or drop: moving right lets G1 support G2 so it can carry B1 left, and returning lets G1 support G3 so that B1 can make contact with BF1. This back-and-forth sequence cannot be eliminated.

Theorems & Definitions (16)

  • theorem 1
  • lemma 1: One-color Red Bend
  • proof
  • lemma 2: One-color OR Gadget
  • proof
  • lemma 3: One-color AND Gadget
  • proof
  • proof : Proof of Theorem \ref{['theorem:pspace-completeness']}.
  • theorem 2
  • proof
  • ...and 6 more