Permutation Match Puzzles: How Young Tanvi Learned About Computational Complexity

2026-03-09

AI Summary

This study investigates the solvability and repair of permutation-matching puzzles on an $n \times n$ grid subject to row- and column-wise sorting constraints (either ascending or descending). By constructing a constraint graph, the work provides the first complete characterization of solvability conditions, introducing a “at most one switch” criterion to determine the existence of a solution. For solvable instances, it presents a counting method based on the hook-length formula; for unsolvable ones, it designs a linear-time algorithm to compute the minimum number of label flips required for repair. The framework is further extended to arbitrary permutation constraints, where the minimum repair problem is shown to be NP-complete. Integrating combinatorics, graph theory, and computational complexity, this work establishes a theoretical foundation and efficient algorithmic tools for this class of puzzles.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Security and Privacy: Large-scale security measurements

Abstract

We study a family of sorting match puzzles on grids, which we call permutation match puzzles. In this puzzle, each row and column of a $n \times n$ grid is labeled with an ordering constraint -- ascending (A) or descending (D) -- and the goal is to fill the grid with the numbers 1 through $n^2$ such that each row and column respects its constraint. We provide a complete characterization of solvable puzzles: a puzzle admits a solution if and only if its associated constraint graph is acyclic, which translates to a simple"at most one switch"condition on the A/D labels. When solutions exist, we show that their count is given by a hook length formula. For unsolvable puzzles, we present an $O(n)$ algorithm to compute the minimum number of label flips required to reach a solvable configuration. Finally, we consider a generalization where rows and columns may specify arbitrary permutations rather than simple orderings, and establish that finding minimal repairs in this setting is NP-complete by a reduction from feedback arc set.

Problem

• permutation match puzzles
• sorting constraints
• computational complexity
• NP-completeness
• grid puzzles

Innovation

• Permutation Match Puzzles
• constraint graph acyclicity
• hook length formula
• minimum label flips
• NP-completeness