same color triangle (2 player game)

Question

Choose 6 points on a plane, with no three of them collinear. There are 2 players, one with a red pen and the other with a blue pen. The first player selects two points and connects them with a line segment. The second player then selects two points and connects them with a line segment as well.

This process is repeated until one player forms a triangle with all three sides of the same color. This player loses.

Is there a winning strategy for player 1 or player 2? If so, what is it?

Answer 1

I find this game called "Sim (pencil game)"

second player wins

Ramsey theory can also be used to show that no game of Sim can end in a tie. Specifically, since the Ramsey number R(3,3)=6, any two-coloring of the complete graph on 6 vertices (K6) must contain a monochromatic triangle, and therefore is not a tied position. This will also apply to any super-graph of K6. For another proof that there must eventually be a triangle of either color, see the Theorem on friends and strangers.

Computer search has verified that the second player can win Sim with perfect play, but finding a perfect strategy that humans can easily memorize is an open problem.

The game of Sim is one example of a Ramsey game. Other Ramsey games are possible. For instance, the players can be allowed to color more than one line during their turns. Another Ramsey game similar to Sim and related to Ramsey number R(4,4)=18, which again cannot end in a tie, is played on 18 vertices and the 153 edges between them. The two players must avoid to color a monochromatic tetrahedron (a three-dimensional pyramid with four triangular faces).

The Ramsey number R(3,3,3)=17 implies that any three-coloring of the complete graph on 17 vertices must contain a monochromatic triangle. A corresponding Ramsey game uses pencils of three colors. One approach can have three players compete, while another would allow two players to alternately select any of the three colors to paint an edge of the graph, until a player loses by completing a monochromatic triangle. Finding perfect winning strategies for these variants is most likely out of reach.

A technical report1 by Wolfgang Slany is available online, with many references to literature on Sim, going back to the game's introduction by Gustavus Simmons in 1969,[2] including proofs and estimates of the difficulty as well as computational complexity of Sim and other Ramsey games.

References: "wikipedia"

here is some strategy to win:enter link description here

Answer 2

General background

It's a strong positional game under misère play: players choose elements of the set of 15 line segments ("edges") to claim, and the first to have claimed the elements of one of the 20 triangles loses. As noted in the comments, if it weren't for the misère play convention, a strategy-stealing argument would ensure that the first player can win.

As mentioned in the comments, Ramsey Theory (or the "theorem on friends and strangers") ensures that someone will make a monochromatic triangle, so there are no draws. Specifically, since players choose edges of a complete graph and are trying (not) to form monochromatic cliques of a given size, this could be called a clique game, except that it's essentially being played as an avoider-enforcer game (note that since a triangle will be made by someone, both players are acting as both avoider and enforcer).

Answer

This is actually a famous game known as Sim or HEXI. In Graph Ramsey games by Wolfgang Slany, things are intelligently reduced by symmetry and the like to 2309 compressed positions, which are then analyzed by a computer to find that this is actually a win for player 2 under perfect play. (Also it's confirmed that player 2 still wins if players can claim multiple segments at once.)

As far as I and wikipedia know, no human has learned perfect play.

Edit:

It seems wikipedia may be wrong, if the strategy described in "The Game of Sim: A Winning Strategy for the Second Player" by Mead, Rosa, and Huang is correct and not hard to implement. But that paper is from 1974 so I'm a little concerned as to why it was missed. In "Graph Ramsey games", Dr. Slany contends that the strategy in that paper is not easy to memorize for human players. Thanks to Soheil0098's self-answer for pointing out that paper.