Permutation Match Puzzles: How Young Tanvi Learned About Computational Complexity

Gajjar, Kshitij; Misra, Neeldhara

doi:10.4230/LIPIcs.FUN.2026.20

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LIPIcs.FUN.2026.20.pdf

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Document Identifiers

DOI: 10.4230/LIPIcs.FUN.2026.20
URN: urn:nbn:de:0030-drops-257398

Subject Classification

ACM Subject Classification

Theory of computation → Design and analysis of algorithms
Mathematics of computing → Combinatorics

Keywords

sorting match puzzles
permutation match puzzles
grid puzzles
constraint satisfaction
directed acyclic graphs
hook length formula
standard Young tableaux
NP-completeness
feedback arc set

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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 × 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 n² 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.

Cite As Get BibTex

Kshitij Gajjar and Neeldhara Misra. Permutation Match Puzzles: How Young Tanvi Learned About Computational Complexity. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.20

Author Details

Kshitij Gajjar

IIIT Hyderabad, India

Neeldhara Misra

IIT Gandhinagar, India

Acknowledgements

We would like to thank the Center for Creative Learning (https://ccl.iitgn.ac.in/) at IIT Gandhinagar for introducing us to this puzzle. We also acknowledge the use of an AI assistant (https://claude.ai/) for the preparation of this draft. We are also grateful to the reviewers of FUN 2026 for their feedback.

References

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