The Fewest Clues Problem

Authors Erik D. Demaine, Fermi Ma, Ariel Schvartzman, Erik Waingarten, Scott Aaronson

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Erik D. Demaine
Fermi Ma
Ariel Schvartzman
Erik Waingarten
Scott Aaronson

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Erik D. Demaine, Fermi Ma, Ariel Schvartzman, Erik Waingarten, and Scott Aaronson. The Fewest Clues Problem. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 12:1-12:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


When analyzing the computational complexity of well-known puzzles, most papers consider the algorithmic challenge of solving a given instance of (a generalized form of) the puzzle. We take a different approach by analyzing the computational complexity of designing a "good" puzzle. We assume a puzzle maker designs part of an instance, but before publishing it, wants to ensure that the puzzle has a unique solution. Given a puzzle, we introduce the FCP (fewest clues problem) version of the problem: Given an instance to a puzzle, what is the minimum number of clues we must add in order to make the instance uniquely solvable? We analyze this question for the Nikoli puzzles Sudoku, Shakashaka, and Akari. Solving these puzzles is NP-complete, and we show their FCP versions are Sigma_2^P-complete. Along the way, we show that the FCP versions of 3SAT, 1-in-3SAT, Triangle Partition, Planar 3SAT, and Latin Square are all Sigma_2^P-complete. We show that even problems in P have difficult FCP versions, sometimes even Sigma_2^P-complete, though "closed under cluing" problems are in the (presumably) smaller class NP; for example, FCP 2SAT is NP-complete.
  • computational complexity
  • pencil-and-paper puzzles
  • hardness reductions


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  1. Charles J Colbourn. The complexity of completing partial Latin squares. Discrete Applied Mathematics, 8(1):25-30, 1984. Google Scholar
  2. Erik D Demaine, Yoshio Okamoto, Ryuhei Uehara, and Uno Yushi. Computational complexity and an integer programming model of shakashaka. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 97(6):1213-1219, 2014. Google Scholar
  3. Ian Holyer. The NP-completeness of some edge-partition problems. SIAM Journal on Computing, 10(4):713-717, 1981. Google Scholar
  4. David Lichtenstein. Planar formulae and their uses. SIAM Journal on Computing, 11(2):329-343, 1982. URL:
  5. Brandon McPhail. Light up is NP-complete. Unpublished manuscript, 2005. Google Scholar
  6. Wolfgang Mulzer and Günter Rote. Minimum-weight triangulation is NP-hard. CoRR, abs/cs/0601002, 2006. URL:
  7. Takahiro Seta. The complexities of puzzles, cross sum and their another solution problems (ASP). Senior thesis, Univ. of Tokyo, Dept. of Information Science, Tokyo, Japan, Feb 2001. URL:
  8. Yato Takayuki and Seta Takahiro. Complexity and completeness of finding another solution and its application to puzzles. IEICE transactions on fundamentals of electronics, communications and computer sciences, 86(5):1052-1060, 2003. Google Scholar
  9. Seinosuke Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865-877, October 1991. URL:
  10. Nobuhisa Ueda and Tadaaki Nagao. NP-completeness results for nonogram via parsimonious reductions. preprint, 1996. Google Scholar
  11. Leslie G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189-201, 1979. URL:
  12. Leslie G. Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3):410-421, 1979. URL:
  13. Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(C):85-93, 1986. URL: