License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FUN.2018.3
URN: urn:nbn:de:0030-drops-87944
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Abel, Zachary ; Bosboom, Jeffrey ; Demaine, Erik D. ; Hamilton, Linus ; Hesterberg, Adam ; Kopinsky, Justin ; Lynch, Jayson ; Rudoy, Mikhail

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

LIPIcs-FUN-2018-3.pdf (0.8 MB)


We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding Sigma_2-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies.

BibTeX - Entry

  author =	{Zachary Abel and Jeffrey Bosboom and Erik D. Demaine and Linus Hamilton and Adam Hesterberg and Justin Kopinsky and Jayson Lynch and Mikhail Rudoy},
  title =	{{Who witnesses The Witnessl Finding witnesses in The Witness is hard and sometimes impossible}},
  booktitle =	{9th International Conference on Fun with Algorithms (FUN 2018)},
  pages =	{3:1--3:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-067-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{100},
  editor =	{Hiro Ito and Stefano Leonardi and Linda Pagli and Giuseppe Prencipe},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-87944},
  doi =		{10.4230/LIPIcs.FUN.2018.3},
  annote =	{Keywords: video games, puzzles, hardness}

Keywords: video games, puzzles, hardness
Collection: 9th International Conference on Fun with Algorithms (FUN 2018)
Issue Date: 2018
Date of publication: 04.06.2018

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