Use the given clues to solve the puzzle below.
Note: an image editing tool is recommended.
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1$\begingroup$ Intriguing. I'm not having time at the moment, but I like it already ;c) $\endgroup$– BmyGuestFeb 22, 2015 at 21:29
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$\begingroup$ Is the 'solution' a final, single image, or is it a sequence of images? The 2nd positive clue seems to suggest a step-wise 'increasing' solution rather than a static moment. Also: Can the same logical behaviour be assumed in all 4 directions from a node? (Otherwise there might not be enough clues.) $\endgroup$– BmyGuestFeb 22, 2015 at 21:36
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$\begingroup$ The solution is a final image; the 2nd positive clue just indicates what happens if you make a connection between two nodes. Yes, the same logical behavior applies to all four directions, although that could be determined by assuming so. =) $\endgroup$– Bulldogg6404Feb 22, 2015 at 21:41
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$\begingroup$ Are we allowed to post partial solutions, or do you prefer people come up with the final solution only? I'm having some ideas what the clues may mean, but am not certain on some aspects. $\endgroup$– BmyGuestFeb 22, 2015 at 21:42
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$\begingroup$ You may do as you please. I not only enjoy knowing people can solve the puzzle, but I thoroughly enjoy seeing the thought processes people go through when solving. To me, it's like hearing the story of their journey. $\endgroup$– Bulldogg6404Feb 22, 2015 at 21:49
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2 Answers
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Solution
Explanation
The positive and negative clues are supposed to mean:
We need to connect points of the same colour with straight lines such that if a point is marked N, N lines should come out of it. Black means an unknown colour - it can be connected to a point of any colour, but it will then have to be coloured accordingly. It is thus not allowed to connect points of different colours to a black point. Connecting lines should not cross at anywhere except numbered points Points of the same colour should form a connected component
This puzzle is just a variant of Bridges, which you can play here for example.
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1$\begingroup$ I really like that the four sets of bridges have the same shapes, just rotated. There's nothing in the problem that requires that to be the case, so it doesn't needlessly make the problem easier, but it's very satisfying to notice when the puzzle is solved. $\endgroup$– KevinFeb 23, 2015 at 0:05
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$\begingroup$ Something to note from the start, however, is that the grid has rotational symmetry in all four quadrants. True, it isn't required, but it is within reason for them to be the same shape in theory. $\endgroup$ Feb 23, 2015 at 3:24
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This is work in progress.
Following the dedicated invitation of Bulldogg6402, I'm going to chronologically report my thought-process, vowing to not read other solutions (until one becomes accepted at least). The following text is therefore not a well-thought 'summary' but rather a 'storyline' of thoughts where later statements might contradict earlier ones. And there might be plenty of dead-ends.
Why?
Because this site is about learning. Learning both to solve and to build puzzles. By giving my full thought-process, I'm giving the puzzle-author the ability to refine his puzzle or understand how it works 'on others', and I might also give other puzzle-solvers some ideas on how one can tackle a puzzle (or fail at it.)
The first part of this puzzle is obviously to understand the clues, so I'm trying to make sense of those first.
What does positive and negative clue mean?
I assume that by 'negative' clues situation are illustrated which are forbidden, whereas positive clues show valid/wanted situations.
This is hardened by the fact that he respective first clue images are the opposite of each other as far as bridges are concerned.
What do the numbers mean?
First there, is the observation that only 1, 2 and 3 exist, so there are two possible "natural" meanings:
A) - The numbers represent order of actions: First build bridges from 1-nodes etc.
B) - The numbers represent connectivity: A "2" node has 2 bridges etc.
Lets investigate the clues for both options:
For option B)
The last negative clue would fit. The "1" node is forbidden to have 2 bridges. The second negative clue would be invalid, but there is also the matter of colours, so it might still fit. However, the positive clues are at least incomplete as there are two "2" nodes with only one bridge shown.
For option A)
The first positive clue would then be interpreted as the "1" connects to the "2" and then further to the second "2". This second 'connection' is not straight forward logical, though. The negative clues are also odd. The forth would indicate that one must not connect to two different other nodes starting from a "1" and that it is forbidden to go from "1" to another "1". (Which is in contrast to the positive "1" connects to "2" connects further to "2" interpretation.)
It seems to be, that option A is the more logical conclusion. So lets look on the board with this assumption, because there is an immediate problem:
Some "2" and "3" nodes seem to be too isolated to have the according number of connected bridges, in particular if the first negative clue indicates that 'empty' nodes must not be connected.
What do the colours mean?
From the first positive clue, it seems to indicate some sort of 'priority'. If there is a black bridge connecting a colour-node, it colorizes bridge and follow-up node. However, there are no clues (neither positive nor negative), which of the 4 non-black colours have priority, if they touch. This seems to be a problem in particular in the central region. Also, the process of "colourizing" a node seems to indicate towards number option A) not B).
END OF INITIAL THOUGHTS
(Lack of time and going to bed now ;c) )
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$\begingroup$ Good analyses! I'm surprised, because I didn't expect that the numbers on the nodes would be interpreted as an order. I didn't put anything in my clues that would have guaranteed against it, so I have to concede that I didn't make it entirely clear at the start. Thanks! $\endgroup$ Feb 23, 2015 at 0:20
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$\begingroup$ Nice puzzle. As for the clues: I still don't get the meaning of the negative clue with the two black 1's. Also: I would have included a clue making it clear that a bridge may exceed the length of one. $\endgroup$– BmyGuestFeb 23, 2015 at 6:29
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1$\begingroup$ Connecting two 1's would normally be legal, being one connection per node. However, it would isolate the pair from the rest of the nodes, which is not intended - so I added it as a contradiction, a useful rule for solving it. I intended to make a rule that showed a bridge could be longer than one unit, but a handful of nodes are more than a single unit away from the other nodes anyway, which made it a rule that could be learned by practice rather than it being given. It wouldn't have been much of a puzzle to explain everything before you start, would it? =) $\endgroup$ Feb 23, 2015 at 7:00
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$\begingroup$ Fair points, both. A shame that the time difference came between me and the puzzle.;-) good though: just seeing your puzzle gave me a new idea for one myself. Not sure when I ll have the time to build it though... $\endgroup$– BmyGuestFeb 23, 2015 at 7:05
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$\begingroup$ Yes, that is unfortunate. I posted this puzzle at mid-day, for me, so if I am able to post anything new in the morning or the evening, you might get more time to see it. $\endgroup$ Feb 23, 2015 at 7:23