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24

I claim that The proof:


18

I believe this is the unique solution:


15

Up to symmetries of the board, there aren't very many possible moves for the first player: Does this strategy work?


15

Partial answer: solved the Masyu puzzle. (My first time solving one of these!) Step-by-step solution: The letters not covered by the path spell: Argh! It seems this puzzle just got bigger. Here's the new version: Solution to that one: This time the unused letters spell out


13

I believe this is the unique solution: Key steps: My apologies for the inelegance of my diagram; as you may guess, I drew some things in grey before their colours were resolved.


12

By adding a we get


12

I'd label the 4 subgrids and the circles first. (Apparently 24px font size isn't clear(big?) enough here. Oh well.)


11

First of all, let's adopt the convention that the path runs "macroscopically" anticlockwise. Now, Now, in a Total Masyu puzzle, when the path enters an empty square Therefore, The path forms a single loop, so In that case So


10

Completed: Interim steps (mostly obvious I think... comment if anything is unclear and needs some explanation):


9

Solution Explanation When you fold this cube net, you would get this connection: Start from the obvious one, the two adjacent black cells: Then some more: Consider this: What if you connect the lines below the white cells, in the 6-celled side? (shown in dark green) So the left one needs to be connected on the black cell. Now for the question mark: ...


9

(Unenterable squares here are marked with a ?.) Basic deductions: Extending those pieces: Final answer:


9

There are two solutions for 7x7 grids (not counting reflections and rotations) I found these by brute force. Each grid can be abstracted as a permutation of 1234567, where each digit represents the column a black pearl is placed on the row correlating to the digit's position. I used this javascript page to generate a list of all 5040 permutations in a ...


9

Solved! I use X, Y, and Z as axes to describe my logic.


9

I feel a bit guilty posting this as an answer since I know that at least StephenTG had done most of it before and also because frankly Rand did most of the work. But here we go. After Rand's solving we have and now Note:


8

Deusovi came up with a very nice and simple solution. Here is the (much more convoluted) configuration that I came up with, if you were curious: The line:


8

The solution is Explanation:


8

Differences in 3D Starting with the white circles and slowly expanding to all other circles gives Now we have to check for possible ways to connect the segments Which leads to the solutions


7

8x8 has been solved Give it a go! Yes, I know what shape that makes >:)


7

Just a note that I think there's a slightly different solution to Deusovi's:


7

I hope I understood the instructions correctly. If so, the correct answer is:


7

Haven't done these before, but I think I have a solution:


6

Observations Finished grid


5

I'm too lazy to add an explanation?


5

Some grids for question 1: 44 circles 45 circles 45 circles


4

[EDITED to add:] Oops, this is not actually an answer to the question as posed because I failed to notice the requirement that there be a unique solution. I have now verified that my surely-non-optimal answer to Q1 is in fact a uniquely solvable puzzle. I haven't given any thought to that question in regard to Q2 yet. It might be rather hard. A surely non-...


4

Partial answer - stuck on extraction Solution to the Nurikabe: What else is going on? I'm not sure what to do next; overlaying the solutions to the logic puzzles doesn't look like anything useful. (And the fact that the 4×4 doesn't appear to work is surprising...)


4

By removing We can generate the following uniquely solvable solution Reasoning


4

There you go: Anyone can use this for the playing board.


3

There are actually 3 solutions


3

With thanks to @Deusovi for solving the Masyus and doing the hard part. If you overlay the Masyu solutions on the Nurikabe solution, you will find something interesting: You notice that To extract: So to make this Masyu more colorful, you should probably do this: There, much better.


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