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24

I claim that The proof:


15

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


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.


11

By adding a we get


10

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


9

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


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: ...


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

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

By removing We can generate the following uniquely solvable solution Reasoning


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

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


3

There are actually 3 solutions


3

Partial answer The Russian desman After much staring and thinking and dictionary-trawling I remain confused by 8d, which I can't make any sense of either as a p.d. or as an ordinary clue. [EDITED to add: M Oehm figured it out; see below.] Grids: Explanations: Credit where due:


2

I've found one solution, through some logical reasoning followed by trial and error. Thanks to Skynet_0 for pointing out that one of my solutions is invalid.


2

I agree with @w l on part 1 of this question, we all seemed to get the exact same pattern. Part 2 is still open, so here's my take: Also


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