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I have a partially played game of sprouts that I need help solving. The rules are simple (Kevin over at VSauce2 did a great job explaining it if you don't like reading, though the Yoda voice is a bit over the top):

  • Each player takes turns drawing a line from one dot to another.
    • After drawing a line, a new dot is placed anywhere on the line so long as it doesn't intersect with any other dot.
  • A dot can not intersect with more than three lines.
    • A dot with three intersecting lines is considered dead.
  • Lines can:
    • Be straight.
    • Can curve.
    • Can form complete circles back to the starting point.
    • Can never intersect other lines.

The object of the game is to force your opponent into having no playable moves remaining. This is accomplished when all dots are either dead or unplayable based on the rules above. For example with a two dot starting point and red starts, and red wins:

Start Start + 1 Start + 2 Start + 3 Start + 4 Final

Now that you know how to play, I've found an old game of sprouts laying around and need help figuring out the best strategy to win. Red moves next.

Sprouts

If both players are making the most advantageous move for themselves:

  • What is the fewest number of moves possible to complete the game?
    • Who wins in this scenario?

This is a golf style puzzle meaning the answer with the fewest number of moves wins.

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First, let's hand out names for living dots. The free-standing dot with no lines is A. The dot to the lower right is B, the other dot that can be reached from A is C, and the one dot that can only be reached from C is D

The remaining moves are either 2 or 4, depending on how they are made. If A connects to both B and C (2), that'll open up two more dots, which can connect to one another (3), creating a dot that can connect back to the one unfilled slot on A (4). (A slight variant occurs if A connects to either B or C, and then connects the resulting dot to the other one, but it still comes out at 4.) If D connects to C, that leaves only B to connect to A, leaving 2 connections made. Likewise, if B connects to C, only the newly generated dot can be connected to A, leaving two connections. Red wins, regardless of the strategy that either player might employ.

Or, in short

fewest moves: 2. Red wins.

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