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I have invented a new puzzle called Intersection. Let's find out what it is!


Intersection

You are given an $n\times n$ grid with circles and boxes inside arbitrary squares. A circle cannot be in the same grid square as another box; there must always be at least two circles, but at least one box; the circles and/or boxes cannot fill up every square in the grid. Now onto the real rules of the game!

Suppose we let $n=3$ and have the following configuration:

$$\begin{array}{|r|c|} \hline & & \\ \hline &\bigcirc &\square \\ \hline \bigcirc & & \\ \hline \end{array}$$

The aim of the game in this case is to connect the two circles with lines, but the lines must intersect at the grid square with the box in the middle.

  • Lines only start from circles, not boxes or anywhere else;
  • When a line reaches an outer edge of the grid from a particular grid square, the line can be continued from the grid square on the directly opposite end of the same row/column, unless a line or circle is in the grid square on that very opposite end (kind of like how Pac-Man can leave the maze from one end and enter it back in from the directly opposite end);
  • Lines cannot start from a circle and then connect back to the same circle;
  • Lines cannot interfere with each other in regular grid squares, but only on the squares where the boxes are in;
  • Lines can only travel in rows and columns, no diagonals;
  • Up to four lines can protrude from a given circle in general, but that can vary depending on the position of a circle.
  • Lines can only connect to other circles by crossing at least one grid square — if two circles are in grid squares adjacent to each other, a line cannot connect them in between.
  • Lines must fill every grid square!

With those rules, here is the solution (though there could be more than one, but that I don't know for sure):

$\qquad\qquad\qquad\qquad\qquad\quad$Solution

How about something else?

$$\begin{array}{|r|c|} \hline & &\bigcirc \\ \hline &\bigcirc & \\ \hline \square & & \\ \hline \end{array}$$

The solution to this is:

$\qquad\qquad\qquad\qquad\qquad\qquad\quad$Solution 2
(sorry for the bad drawing skill; trust me, I'm better by hand than by software, but that's besides the point)

So, let's bump it up to something a little harder, shall we?


Puzzle

Solve the following intersection grid!

$$\begin{array}{|r|c|} \hline \bigcirc& & & \\ \hline & &\bigcirc & \\ \hline &\square &\square & \\ \hline & &\bigcirc &\square \\ \hline \end{array}$$

The first user to answer with a solution will get the tick; another answer to follow that might also have a solution will get a $+50$ rep bounty; if an answer holds more than one solution, then it will get a $+100$ rep bounty (and if that one comes first, it will instead get the tick).

I hope my puzzle makes sense.

Good luck! :D


P.S. This game is like Flow for those who know that game, except the lines, well, intersect each other.

Edit: Turns out there is a game called Flow Free: Bridges such that lines can intersect each other. Unfortunately, this was unbeknownst to me when I created and posted this puzzle. My apologies.

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    $\begingroup$ Must we fill in every square? Asking this given the "Flow" reference. $\endgroup$ – EKons Apr 3 at 21:08
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    $\begingroup$ This is like the sequel Flow Free: Bridges. $\endgroup$ – noedne Apr 3 at 21:09
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    $\begingroup$ I don't think it's the exact concept, since we have to connect all pairs of circles, not just each pair of same-colored circles (your grid has three such pairs, for example, Flow Free: Bridges needs an even number of circles because of the way it works). $\endgroup$ – EKons Apr 3 at 21:14
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    $\begingroup$ @user477343 You say, "This game is like Flow except the lines intersect each other." In Flow Free: Bridges, there are bridges like your squares where the lines can intersect. $\endgroup$ – noedne Apr 3 at 21:21
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    $\begingroup$ Please don't add irrelevant tags to your questions; tags have "tag wiki" pages where you can see what tags are used for. $\endgroup$ – Deusovi Apr 4 at 2:20
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solution­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­

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  • $\begingroup$ Yes, this was the solution I had! Good job! $(+1)$ I will give you the tick in $24$ hours from now! :D $\endgroup$ – Feeds Apr 4 at 1:47
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I think I've got an alternative solution to noedne

enter image description here

Here are two more solutions if we include the rule that at least two lines must protrude from a circle

enter image description here enter image description here

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    $\begingroup$ Hmm, if I say that at least two lines must protrude from a circle, then this narrows down multiple solutions, which would exclude your answer and make @noedne's answer correct. I'll make sure to include that rule if I show this puzzle to some of my friends not on this site; but for now, I accept this as an alternative solution and will give you a $+50$ rep bounty (though I will have to wait two days form now). Well done! :P $(+1)$ $\endgroup$ – Feeds Apr 4 at 1:44
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    $\begingroup$ @user477343 Adding some restrictions is probably a good idea. I like the idea in this puzzle and it would be cool to see how much it can develop. I've added what I think are two more solutions based on the new restriction. $\endgroup$ – hexomino Apr 4 at 9:04
  • $\begingroup$ Congratulations! $\boxed{\color{blue}{+50}}$ $\endgroup$ – Feeds Apr 7 at 11:47
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It's not too different to the above, but I think this counts as

Another solution:

An Intersection Solution

And here is one that is different

Another solution:

An Intersection Solution 2

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    $\begingroup$ As far as I can tell, rule 2 seems to forbid a line from wrapping around the grid if there is a circle in the next cell. So, for example, I think the line going up to the bottom circle is disallowed. $\endgroup$ – hexomino Apr 5 at 12:48
  • $\begingroup$ @hexomino indeed! $\endgroup$ – Feeds Apr 5 at 23:21
  • $\begingroup$ Your answer is unfortunately not a solution because of Rule 2, as previously mentioned by @hexomino: When a line reaches an outer edge of the grid from a particular grid square, the line can be continued from the grid square on the directly opposite end of the same row/column, unless a line or circle is in the grid square on that very opposite end. Sorry about that! $\endgroup$ – Feeds Apr 5 at 23:22
  • $\begingroup$ Oh, right. That is a weird rule. Whoops. $\endgroup$ – Elements in Space Apr 6 at 1:12
  • $\begingroup$ Such rules are added to increase the difficulty of the puzzle. Rule 2 in particular, however, was also included because lines can only connect via full squares. If we include this rule and then have two circles on opposite ends of the same row/column that we wish to connect, this wish would violate that rule. Not allowing such a position as opposed to changing the rule is more complicated, however (in my opinion). (+1) for your effort, nevertheless! :) $\endgroup$ – Feeds Apr 6 at 2:21

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