This is a problem I encountered in this page:
Given an N-sized grids, like the figure A shown below (as N = 4). The blue points are the places the first player can choose, and the red points are the places the second player can choose.
In the game, the two players take turns to choose two points to get connected by a stick. The two chosen points’ distance should be exactly one-unit length. The first player’s goal is to create a ‘bridge’ that connects a most left point and a most right point. The second player’s goal is to create a ‘bridge’ that connects a most top point and a most bottom point. Figure B shows a possible result (the first player won). In addition, the stick shouldn’t get crossed.
It is said that the first player has a winning strategy, since the grids are symmetric and whoever moves first has an advantage. However, I don't think that the explanation is mathematically rigorous. I need an explanation that is logical and convincing. The explanation can be either constructive or nonconstructive.