# Hotaru Beam Puzzle: Is it impossible?

I saw this puzzle on Twitter by user @lilva_0419 and I personally feel that it is impossible to solve. Can you prove me wrong (or right)?

Try it online

Rules (see Wikipedia):

1. Connect every white circle with lines into a single network.
2. Lines start at black dots and end on white circles (not black dots!)
3. Lines can make 90 degree turns, but cannot intersect or branch off of each other.
4. The number in the circles gives the number of turns the line (starting at this circle's black dot) makes before reaching a white circle.

[Begin Edit] Here is the actual wording of the rules from Wikipedia. This is just for future reference, the accepted answer is correct, but this should clarify why the creative "think-outside-the-box" answer by FibS would unfortunately not work here. I apologize for my unfortunate wording.:

1. Draw a line from each white circle's black dot to another white circle, following the grid's horizontal and vertical markings.
2. Lines cannot be drawn from a black dot to another black dot, nor can they be drawn from a white circle not at its black dot to a white circle not at its black dot.
3. No crossing or branching of lines is acceptable. At the end, the drawn lines will connect all white circles to form a single, contiguous network.
4. The number on the white circle dictates how many times the line you draw '''from its black dot''' must bend before it meets another white circle. [End Edit]

[Begin 2nd edit] Apparently the Wikipedia translation of the first rule now reads (emphasis mine):

Draw a line from each white circle's black dot to any white circle, following the grid's horizontal and vertical markings.

Thanks @edderiofer for pointing it out.

That further supports the accepted solution.

[End 2nd edit]

I see two options for me being wrong: I misunderstood/forgot a rule. Or I overlooked a possibility in the puzzle.

The reason I think this is impossible:

In the top left corner the two turn line can have its first turn after one or two steps. After two steps it can connect to multiple other circles with another turn, but it cuts off the blank circle in the top row (see Figure 1). Turning after just one step only leaves one possible connection through another turn: The left of the two circles in the third row from the top. However that needs to connect to the blank circle in the top row, which leaves no reachable circle for that very same blank circle to connect to (see Figure 2).

Figure 1: Part I of the reason I suspect the puzzle has no solution.

Figure 2: Part II of the reason I suspect the puzzle has no solution.

• "Lines can make 90 degree turns" This does not say that the lines must align with the grid. Jan 21, 2021 at 19:35
• How could 2 Lines start at black dots and end on white circles (not black dots!) ever work, please? "… white circles (not black dots!)…" is hardly clear but what could it mean if not "… white circles without black dots…"? Jan 21, 2021 at 21:50
• @RobbieGoodwin, it means "Lines start at circles from the black dot and end on circles (but not at the black dot on the circle!)". Does that help?
– LHM
Jan 21, 2021 at 21:56
• The Wikipedia article has since been corrected to read "any white circle" instead of "another white circle". Presumably from someone here. Jan 23, 2021 at 2:13
• Thank you @edderiofer I added that information to the question.
– niak
Jan 25, 2021 at 9:49

Well

there is no rule that states a line cannot end up on the same circle it started from

so you could do

Full solution

(The "Try it online" link agrees that this is legal)

• I think this is the crux of it; can a line return to its own starting circle? Wikipedia rules say "Draw a line from each white circle's black dot to another white circle, following the grid's horizontal and vertical markings" (emphasis mine) which says to me that a line cannot return to its own circle. OTOH, the online system says it's okay, so... maybe the rules on Wikipedia are wrong? Jan 21, 2021 at 22:19
• These rules say it's okay: janko.at/Raetsel/Hotaru-Beam/index.htm Jan 21, 2021 at 22:22
• @studog Apparently the rules on Wikipedia were translated from this. I don't speak Japanese, but at least popping it into Google Translate says nothing about the same circle. Jan 21, 2021 at 22:24
• A better translation of the first rule is: "From every white circle's black dot, draw a line along the dotted lines to a white circle." There is indeed no mention of it needing to be another circle. Jan 22, 2021 at 6:12
• That is true. I trusted WIkipedia's rules, which sad "another", but of course, Wikipedia is not an official source. Since this puzzle was posted together with the "try it online" link and it works there, I would argue that this is indeed the correct solution. Thanks a lot!
– niak
Jan 22, 2021 at 10:10

Here's a solution that complies with Wikipedia's rules:

There are exactly two lines here, one from the (2) to the (1) and one from the ( ) to the (1). Crucially, the rules state lines cannot cross or branch, but doesn't mention overlapping.

• That might be up for interpretation. My feeling is that allowing a line going straight through another white circle and not ending there would make the entire puzzle less logical and potentially allow for many more solutions. It is true though that making a line longer could be interpreted as not branching off...
– niak
Jan 22, 2021 at 18:11
• @niak I see your concern about branching, a mathematical definition may indeed allow for it. But from my experience, mathematical definitions are rarely used in puzzles, e.g. a "count the trinagles" puzle would always yield the answer "an infinite number".
– Fax
Jan 22, 2021 at 22:40