Game where you construct a line through the board blocking the other player?
The game is called Bridg-It and it is a specific implementation of the "Shannon switching game".  There's a very elegant win for the first player which is based on pairing the edges together.  I think it's in a book by Martin Gardner, but I can't seem to find it online.  There's another solution here, but it's not as easy to understand.

Edit: The solution is given in New Mathematical Diversions by Martin Gardner, and was found by Oliver Gross.  "Make the first play as indicated by the black line at lower left in the diagram. Thereafter whenever your opponent's play  crosses the end of a  dotted line, play by crossing the other end of the same line."