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Based on a puzzle given to us by Jamal Senjaya I created a bigger one. Hope it is not too easy. enter image description here

The number in circle means how many of adjacent triangles should be painted blue.

EDIT
I did not mention (I was not aware) that the puzzle has many (I hope not too many) possible solutions. There is one however that "makes more sense" - if you see it you will know that this is it. All valid solutions will earn an upvote (hopefully not only from me), but only "the one" will be accepted. The desired solution can be distinguished by the fact that the resulting image is a 4-letter word.

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    $\begingroup$ Are you sure there is a unique answer ? The hard part to create this kind of puzzle is to make sure there is only 1 unique answer. $\endgroup$ – Jamal Senjaya Oct 20 '16 at 7:18
  • $\begingroup$ There are LOTS of possible solutions. There are some variations you can make more to the center and all solutions can change a lot of the colors in the outer circle to still produce a valid coloring. So it seems really hard to find which solution you intended. $\endgroup$ – w l Oct 20 '16 at 7:39
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    $\begingroup$ My calculations show 14,233 (!) possible solutions. $\endgroup$ – 2012rcampion Oct 20 '16 at 8:24
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    $\begingroup$ @2012rcampion Care to share them? Ah, there is a separate question for the calculations like this. $\endgroup$ – oleslaw Oct 20 '16 at 8:27
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    $\begingroup$ I used SatisfiabilityCount which only counts the solutions, instead of returning them all. (Getting all of them is much much slower.) $\endgroup$ – 2012rcampion Oct 20 '16 at 8:33
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This displays the missing verbal ingredient, which certainly does add a nice dimension to the puzzle.

$ \small\sf \color{black}{\text{The word is }} \normalsize\it \rlap{NICE} $

Began with Jonathan Allan’s report of which triangles had certain values, then considered the four-letter hint, noticed what the third letter must be, and defocused for the rest.

Arrows point where the goal of forming letters suggested filled-in triangles in the top half. Each suggestion led to a few more forced values, until three fairly-certain letters were apparent.

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    $\begingroup$ Very nice solution. This is "the one" I was thinking about. Happy to see it has been solved. I think you should mention the word explicitly in your answer - just for clarity. $\endgroup$ – oleslaw Oct 20 '16 at 11:34
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    $\begingroup$ Nice how the solution was constrained enough by a mathematical first stage to allow for a nonmathematical second stage. The hint was probably necessary after all. $\endgroup$ – humn Oct 20 '16 at 11:36
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    $\begingroup$ This can't be the right answer since none of the triangles are painted blue. ;-) $\endgroup$ – David Conrad Oct 20 '16 at 14:08
  • $\begingroup$ Oboy, this must've been an answer to some other puzzle. To make up for that, more than half of the triangles are now blue. $\endgroup$ – humn Oct 20 '16 at 22:36
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Here is another possible solution, with a first stage that is necessary:

Red means can't be blue in the top image and blue means must be blue, the second image is then one way to fill in the rest. a solution with necessary first step

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  • $\begingroup$ I wish I could upvote +2 for pointing out the "necessary" part. $\endgroup$ – oleslaw Oct 20 '16 at 7:35
  • $\begingroup$ The "necessary" part is the easy one. After that part, you pretty much have to takes guesses due to multiple possibilities. $\endgroup$ – stack reader Oct 20 '16 at 7:37
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    $\begingroup$ @stackreader Next puzzle - how many solutions does this Hexagonal Minesweeper™ have? $\endgroup$ – Jonathan Allan Oct 20 '16 at 7:40
  • $\begingroup$ @JonathanAllan That would be fun actually.... but might nmke my brain hurt. $\endgroup$ – stack reader Oct 20 '16 at 7:41
  • $\begingroup$ @ Jonathan Allan Just did it $\endgroup$ – oleslaw Oct 20 '16 at 7:42
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I feel like there might be many possibilities but here is mine.

enter image description here

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  • $\begingroup$ Also no blue here ;-) $\endgroup$ – Paŭlo Ebermann Oct 20 '16 at 19:09
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Here's my version.

Solved Puzzle

I actually started at the top with the two 5 nodes and just worked my way around. I did the 0 nodes before messing with the surrounding nodes, then did the 1 nodes, then the other 5 nodes. Everything else was kind of random, but I didn't have to re-do anything.

Blue are "blue" nodes. I used gray to say "this can't be a node" as I went along, then left it that way when I finished.

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I made a little program to demonstrate this puzzle. It is an interactive thing that you can click on and off to select "mines". I made blue the color of the safe spots, and red the color of mines. White is simply unselected:

My solution:

enter image description here It is the only spots I can prove are filled in that manner.

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