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An entry in Fortnightly Topic Challenge #32: Grid Deduction Hybrids


I received an envelope containing two drawings and no return address. I would recognize that handwriting anywhere - my dear friend has written me a letter! It would appear that wherever he was, he's okay now. Where he is now is another question...

I am an Island 1

I am an Island 2

No Levels correction: Image 1

No Levels correction: Image 2

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    $\begingroup$ Did I already tell you, that I very much like your visual style? :c) $\endgroup$ – BmyGuest Jun 30 '17 at 6:06
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    $\begingroup$ @paramesis First of all, this puzzle was pretty hard and awesome. Secondly, I've finished every one of the grid-deduction puzzles and after solving it I don't really see the point of the middle puzzle in the second picture. Is there something I have to do with its solution or am I done with the whole thing? $\endgroup$ – SirGrapefruit Jul 12 '17 at 0:39
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    $\begingroup$ @SirGrapefruit I am an island. $\endgroup$ – paramesis Jul 12 '17 at 0:42
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    $\begingroup$ @SirGrapefruit no OUTSIDE information needed...you'll find it $\endgroup$ – paramesis Jul 12 '17 at 2:03
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    $\begingroup$ @SirGrapefruit Do you have time now? $\endgroup$ – boboquack Oct 19 '17 at 22:54
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Your friend is in:

Hawaii (thanks @Reinier!)

Grid deduction

Handdrawn solutions (originals for reference, click on the links to see):

Top left
Top right
Bottom left
Bottom right
Centre

Prettified solutions, image by @TheGreatEscaper:

enter image description here

My initial set of rules:

Draw some lines orthogonal to the grid between squares. A square may have 0, 1, 2 or 3 lines emanating from it, in each case it shall contain respectively a black square, a black circle, nothing or a white circle. Lines may not cross the border between two regions. The numbers at the sides of the rows/columns count the number of each feature (circles/black square) in some order, fixed for rows and columns separately for each outer puzzle. No two black squares may be adjacent to each other.

That was close. Upon some solving, I realised that:

Blue regions contain one connected component, purple regions two and orange regions three.

That solves all the outer puzzles, with a couple of tricks to help simplify:

The circles and black squares are counted twice, by row and by column. We can then match up the rows and columns of numbers by their sum. Also, treating the lines as a graph, the circles represent vertices of odd degree, so since a graph has an even total degree (each edge is counted twice), there is an even number of circles. Looking at the sums then allows us to identify which sum is the one of black squares.

For any partition of a puzzle into two sections, if there are an even number of circles in each section, there are an even number of lines connecting the two sections, and similarly for odd. (This is another consequence of a similar degree argument on the graph.)

To solve the centre puzzle:

Notice that the correspondences between rows and columns of numbers in each puzzle give us a unique place within each of the corner 3x3 squares to put each of the circles and the black square. Also, a new rule is that it is forbidden to cross a black line.

Now:

Remembering that ysaujdiolkiun is sudoku mixed with yajilin.

We can:

Solve the central puzzle assuming that the circles and the black squares obey the rules of a sudoku (9 of each with no two of the same in a marked 3x3 box, row or column)

Extraction

@TheGreatEscaper points out that:

The outer puzzles contain phrases: in the initial order of the images (left to right, top to bottom), we have "I'd better quit", "I'm losing sleep", "it's a serious problem" and "I'll need help".

And as @Reinier comments:

By taking the contracted letters, we have i HAd, i Am, it Is, i WIll which anagram to HAWAII!

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    $\begingroup$ The solutions spell out phrases: 'I'd better quit, I'm losing sleep, it's a serious problem, I'll need help' $\endgroup$ – TheGreatEscaper Jan 2 '18 at 13:53
  • $\begingroup$ I was thinking that extraction would be finding the patterns of circles + square that are used in the 9x9 puzzle within the other puzzles, and that each pattern would be part of a certain letter, leading to a 9 letter answer. But it doesn't seem to work. Meanwhile, I'm having good fun solving these puzzles now that I know the rules! $\endgroup$ – TheGreatEscaper Jan 2 '18 at 14:50
  • $\begingroup$ Alright, I'm all caught up! Things to note: there are exactly 54 letters in the four messages, which is 9x9 - 9x3, i.e. the clueless squares in the central puzzle, including the four blanks. Just writing the messages out in standard L to R, U to D fashion leaves the letters 'ESOE' in the blanks, which isn't very promising $\endgroup$ – TheGreatEscaper Jan 3 '18 at 3:18
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    $\begingroup$ I think I might have the answer: in the extracted sentences, some letters are contracted at the apostrophes ("I'd", "I'm", "it's" and "I'll"). The letters that are left out are "ha", "a", "i" and "wi". These anagram to "Hawaii". $\endgroup$ – Reinier Jan 21 at 12:08
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    $\begingroup$ What a journey. $\endgroup$ – Quintec Jan 22 at 3:07
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@boboquack seems to have already solved the majority of the puzzle, so I just want to add some observations:


The first picture of the two seems to be the introductory puzzle. The solution to this puzzle is to fit the three "clusters" in the bottom of the picture into the orange area. If you then fill all empty spaces with black squares and satisfy the condition that no two black squares are next to each other, you get the following unique solution:

solution

Note that this solution also satisfies the sudoku ruleset, at least for the rows and colums.

Furthermore, I noticed something else:

There seems to be a pink patch in the bottom left of the orange area in the first picture. At first I thought it might just have been a mistake while painting the picture, but the fact that it's pink seems to make it deliberate.

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    $\begingroup$ Would you believe me if I said that today is the first day in 6 months in which I actually have time to work on this puzzle? -_- Seems like I was two days too late, since @boboquack basically got exactly as far as I got. I just wanted to add some things he seems to have missed. $\endgroup$ – SirGrapefruit Jan 4 '18 at 16:13
  • $\begingroup$ What bad timing :P I tried fitting the three parts into the first grid, but couldn't find this configuration, so good job on finding. $\endgroup$ – boboquack Feb 8 '18 at 7:25

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