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DOI: 10.4230/LIPIcs.FUN.2026.12

A Bookworm Climbs up the Polynomial Hierarchy: Meta-Restoration Complexity in Arithmetic Puzzles

Authors: Brynmor Chapman, Lily Chung, Erik D. Demaine, Yota Irino, Della Hendrickson, Tonan Kamata, and Ryuhei Uehara

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract

In arithmetic puzzles, a partially specified arithmetic expression must be completed to make the computation valid. Arithmetical restoration puzzles require filling in missing digits, while cryptarithms involve assigning digits to letters. The Japanese term mushikui-zan ("bookwormed arithmetic") commonly refers to arithmetical restorations, where we imagine the missing digits have been eaten by a bookworm. Puzzle creator Yousuke Ikeda proposed a new type of puzzle in which a previously designed bookwormed arithmetic with multiplication - known to have a unique solution - has itself been "bookwormed", that is, partially erased. The goal is to restore the specified blanks so that the resulting bookwormed puzzle again has a unique solution. We further generalize this framework: for each k ≥ 2, we define level-k puzzles as those in which type-k blanks must be filled to make the resulting level-(k{-}1) puzzle uniquely solvable. We study the level-k versions of the Boolean satisfiability problem, and show that they form a hierarchy of Σ^P_k-complete decision problems, tightly matching the levels of the polynomial hierarchy. As applications, we show that the level-k arithmetical restoration problem with multiplication is Σ^P_k-complete, as is the level-k cryptarithm problem. On the positive side, we show that level-2 arithmetical restoration puzzles with addition are solvable in polynomial time.

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Brynmor Chapman, Lily Chung, Erik D. Demaine, Yota Irino, Della Hendrickson, Tonan Kamata, and Ryuhei Uehara. A Bookworm Climbs up the Polynomial Hierarchy: Meta-Restoration Complexity in Arithmetic Puzzles. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chapman_et_al:LIPIcs.FUN.2026.12,
  author =	{Chapman, Brynmor and Chung, Lily and Demaine, Erik D. and Irino, Yota and Hendrickson, Della and Kamata, Tonan and Uehara, Ryuhei},
  title =	{{A Bookworm Climbs up the Polynomial Hierarchy: Meta-Restoration Complexity in Arithmetic Puzzles}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.12},
  URN =		{urn:nbn:de:0030-drops-257311},
  doi =		{10.4230/LIPIcs.FUN.2026.12},
  annote =	{Keywords: arithmetical restoration, cryptarithms, polynomial hierarchy, uniqueness quantifier, puzzle complexity}
}

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DOI: 10.4230/LIPIcs.ITCS.2026.47

Dudeney’s Dissection Is Optimal

Authors: Erik D. Demaine, Tonan Kamata, and Ryuhei Uehara

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

In 1907, Henry Ernest Dudeney posed a puzzle: "cut any equilateral triangle ... into as few pieces as possible that will fit together and form a perfect square" (without overlap, via translation and rotation). Four weeks later, Dudeney demonstrated a beautiful four-piece solution, which today remains perhaps the most famous example of dissection. In this paper (over a century later), we finally solve Dudeney’s puzzle, by proving that the equilateral triangle and square have no common dissection with three or fewer polygonal pieces. We reduce the problem to the analysis of discrete graph structures representing the correspondence between the edges and the vertices of the pieces forming each polygon.

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Erik D. Demaine, Tonan Kamata, and Ryuhei Uehara. Dudeney’s Dissection Is Optimal. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 47:1-47:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{demaine_et_al:LIPIcs.ITCS.2026.47,
  author =	{Demaine, Erik D. and Kamata, Tonan and Uehara, Ryuhei},
  title =	{{Dudeney’s Dissection Is Optimal}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{47:1--47:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.47},
  URN =		{urn:nbn:de:0030-drops-253345},
  doi =		{10.4230/LIPIcs.ITCS.2026.47},
  annote =	{Keywords: Geometric Dissection, Dudeney Dissection, Dissection with Fewest Pieces}
}

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A Bookworm Climbs up the Polynomial Hierarchy: Meta-Restoration Complexity in Arithmetic Puzzles

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Dudeney’s Dissection Is Optimal

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