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Nondango is NP-Complete

Suthee Ruangwises

TL;DR

This work proves that the solvability of Nondango is $NP$-complete by presenting a parsimonious reduction from $1$-in-$3$-SAT$^+$ to Nondango. The reduction encodes variables as columns and clauses as rows, using a comprehensive gadget toolkit to enforce per-variable consistency across clauses and the exactly-one-true constraint per clause. Key contributions include the explicit gadget suite (Enforcing a Black Circle, Enforcing a White Circle, Enforcing Two Circles with Different Colors, and connectivity-skipping mechanisms) and a strategy to fill remaining areas to preserve parsimony. Together, the results place Nondango among the class of $NP$-complete pencil puzzles, with implications for puzzle design and solver performance.

Abstract

Nondango is a pencil puzzle consisting of a rectangular grid partitioned into regions, with some cells containing a white circle. The player has to color some circles black such that every region contains exactly one black circle, and there are no three consecutive circles (horizontally, vertically, or diagonally) having the same color. In this paper, we prove that deciding solvability of a given Nondango puzzle is NP-complete.

Nondango is NP-Complete

TL;DR

This work proves that the solvability of Nondango is -complete by presenting a parsimonious reduction from -in--SAT to Nondango. The reduction encodes variables as columns and clauses as rows, using a comprehensive gadget toolkit to enforce per-variable consistency across clauses and the exactly-one-true constraint per clause. Key contributions include the explicit gadget suite (Enforcing a Black Circle, Enforcing a White Circle, Enforcing Two Circles with Different Colors, and connectivity-skipping mechanisms) and a strategy to fill remaining areas to preserve parsimony. Together, the results place Nondango among the class of -complete pencil puzzles, with implications for puzzle design and solver performance.

Abstract

Nondango is a pencil puzzle consisting of a rectangular grid partitioned into regions, with some cells containing a white circle. The player has to color some circles black such that every region contains exactly one black circle, and there are no three consecutive circles (horizontally, vertically, or diagonally) having the same color. In this paper, we prove that deciding solvability of a given Nondango puzzle is NP-complete.
Paper Structure (10 sections, 1 theorem, 8 figures)

This paper contains 10 sections, 1 theorem, 8 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Idea of the Proof
  4. Reduction
  5. Enforcing a Black Circle
  6. Enforcing a White Circle
  7. Enforcing Two Circles with Different Colors
  8. Connecting Two Consecutive Clause Rows with Common Variables
  9. Skipping a Clause Row
  10. Filling Empty Area

Key Result

Theorem 1

Deciding solvability of a given Nondango instance is NP-complete.

Figures (8)

  • Figure 1: An example of a $6 \times 6$ Nondango puzzle (left) and its solution (right)
  • Figure 2: Basic structure of a Nondango instance transformed from a formula consisting of clauses $C_1=x_1 \vee x_2 \vee x_4$, $C_2=x_2 \vee x_3 \vee x_5$, $C_3=x_3 \vee x_4 \vee x_5$, and $C_4=x_1 \vee x_2 \vee x_5$
  • Figure 3: A gadget for creating a forced white circle (left) and its only solution (right)
  • Figure 4: A gadget for enforcing two circles with different colors (left) and its only two solutions (right), where $\{A,B\}=\{\text{black},\text{white}\}$
  • Figure 5: A gadget connecting two consecutive clause rows, forcing the two variable circles to have the same color
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1