Problem 3043. Fill-a-pix - Solver (basic)

Created by goc3

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An example starting board is shown below on the left with the completed board shown to its right. Each number in the board indicates how many surrounding cells, including itself, are to be filled in. There are (up to) nine total cells associated with each number: four immediately adjacent (up, down, left, and right), four diagonally adjacent (one touching each corner), and the central cell (where the number is located).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Based on this logic, all 0's and 9's are determinant, as all nine cells are either empty or filled, respectively, for these numbers. Also, all 6's on the board edges and 4's in the corners should be completely filled in, as they only involve 6 or 4 cells, respectively.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"-1\"/><w:attr w:name=\"width\" w:val=\"-1\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"-1\"/><w:attr w:name=\"width\" w:val=\"-1\"/><w:attr w:name=\"relationshipId\" w:val=\"rId2\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>This problem deals with basic-logic Fill-a-pix problems; a sample step-by-step solution is found</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://www.conceptispuzzles.com/index.aspx?uri=puzzle/fill-a-pix/techniques\"><w:r><w:t>here</w:t></w:r></w:hyperlink><w:r><w:t>. These problems are basic in that each clue can be solved independent of all other clues, though some looping or iteration over the board will be required to solve all the clues. Basic logic can be summarized in the following two cases, regarding the cells associated with each clue: (1) if the sum of filled-in cells equals the clue, all other associated cells should be marked as empty, and (2) if the sum of empty cells equals the total number of associated cells minus the clue, all other associated cells should be marked as filled in.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For this problem, you will be presented with a board full of clues. Write a function to solve the Fill-a-pix puzzle based on those clues, returning a solution of the same size as the board filled with 1's (filled) and 0's (empty).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>A related problem is</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/3042-fill-a-pix-solution-checker\"><w:r><w:t>Fill-a-pix - Solution Checker</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray 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<a href = "http://www.conceptispuzzles.com/index.aspx?uri=puzzle/fill-a-pix/rules">Fill-a-pix</a> is a logic puzzle game similar to Pic-a-Pix (aka Logic art) and Minesweeper. An example starting board is shown below on the left with the completed board shown to its right. Each number in the board indicates how many surrounding cells, including itself, are to be filled in. There are (up to) nine total cells associated with each number: four immediately adjacent (up, down, left, and right), four diagonally adjacent (one touching each corner), and the central cell (where the number is located).Based on this logic, all 0's and 9's are determinant, as all nine cells are either empty or filled, respectively, for these numbers. Also, all 6's on the board edges and 4's in the corners should be completely filled in, as they only involve 6 or 4 cells, respectively.<img src = "http://www.conceptispuzzles.com/picture/27/1558.gif"><img src = "http://www.conceptispuzzles.com/picture/27/1574.gif">This problem deals with basic-logic Fill-a-pix problems; a sample step-by-step solution is found <a href = "http://www.conceptispuzzles.com/index.aspx?uri=puzzle/fill-a-pix/techniques">here</a>. These problems are basic in that each clue can be solved independent of all other clues, though some looping or iteration over the board will be required to solve all the clues. Basic logic can be summarized in the following two cases, regarding the cells associated with each clue: (1) if the sum of filled-in cells equals the clue, all other associated cells should be marked as empty, and (2) if the sum of empty cells equals the total number of associated cells minus the clue, all other associated cells should be marked as filled in.For this problem, you will be presented with a board full of clues. Write a function to solve the Fill-a-pix puzzle based on those clues, returning a solution of the same size as the board filled with 1's (filled) and 0's (empty).A related problem is <a href = "https://www.mathworks.com/matlabcentral/cody/problems/3042-fill-a-pix-solution-checker">Fill-a-pix - Solution Checker</a>.

<http://www.conceptispuzzles.com/index.aspx?uri=puzzle/fill-a-pix/rules Fill-a-pix> is a logic puzzle game similar to Pic-a-Pix (aka Logic art) and Minesweeper. An example starting board is shown below on the left with the completed board shown to its right. Each number in the board indicates how many surrounding cells, including itself, are to be filled in. There are (up to) nine total cells associated with each number: four immediately adjacent (up, down, left, and right), four diagonally adjacent (one touching each corner), and the central cell (where the number is located).

Based on this logic, all 0's and 9's are determinant, as all nine cells are either empty or filled, respectively, for these numbers. Also, all 6's on the board edges and 4's in the corners should be completely filled in, as they only involve 6 or 4 cells, respectively.

<<http://www.conceptispuzzles.com/picture/27/1558.gif>>

<<http://www.conceptispuzzles.com/picture/27/1574.gif>>

This problem deals with basic-logic Fill-a-pix problems; a sample step-by-step solution is found <http://www.conceptispuzzles.com/index.aspx?uri=puzzle/fill-a-pix/techniques here>. These problems are basic in that each clue can be solved independent of all other clues, though some looping or iteration over the board will be required to solve all the clues. Basic logic can be summarized in the following two cases, regarding the cells associated with each clue: (1) if the sum of filled-in cells equals the clue, all other associated cells should be marked as empty, and (2) if the sum of empty cells equals the total number of associated cells minus the clue, all other associated cells should be marked as filled in.

For this problem, you will be presented with a board full of clues. Write a function to solve the Fill-a-pix puzzle based on those clues, returning a solution of the same size as the board filled with 1's (filled) and 0's (empty).

A related problem is <https://www.mathworks.com/matlabcentral/cody/problems/3042-fill-a-pix-solution-checker Fill-a-pix - Solution Checker>.

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Jean-Marie Sainthillier on 25 Feb 2015

Nan or -1 (test 5) for non clue cells ?

goc3 on 25 Feb 2015

Thanks for the catch. I've updated that test case.

Guillaume on 26 Feb 2015

Now, the biggie. Implement a solver for advanced FAP. I never managed to create one that worked on all problems.

goc3 on 26 Feb 2015

I was working on it last night, but it's tabled for now. I can't quite figure out how to solve the advanced logic portion that propagates along adjacent clues (including diagonal) to solve cells outside of that clue's reach.

Jean-Marie Sainthillier on 27 Feb 2015

Good problem but too difficult for a first contact. Maybe a first basic problem with small boards and/or clear cues (0/9). And after only, a more difficult one.

goc3 on 9 Oct 2016

If anyone is interested more in these games, I've made quite a few of them on Sporcle.com. for the first of four Fill-a-pix (aka, mosaic logic puzzle) playlists.

Rafael S.T. Vieira on 26 Sep 2020

I wouldn't call this problem basic since it is very similar to solving a nonogram, which is np-hard, but it is possible to solve this problem in this current instance with logical thinking only. With more complex boards of this game, it is not enough because it would have more than one possible solution or no solution.

goc3 on 29 Sep 2020

@Rafael S.T. Vieira: while you are correct (along with Jean-Marie) in stating that the problem is not easy, the "basic" indicates, as you rightly mention, that simple logic applied to surrounding cells is all that is needed to solve this sort of puzzle. Of course, there are more complex boards wherein the "basic" logic is insufficient to completely solve them.

goc3 on 29 Sep 2020

It appears that the link I supplied earlier was lost. Here it is again:
https://www.sporcle.com/playlists/goc3/mosaic-logic-puzzles

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Problem 3043. Fill-a-pix - Solver (basic)

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