Given a $6\times 6$ chessboard. The chessboard is filled with $18$ dominos (each domino covers $2$ adjacent squares). Prove that one can find a line from the one side of the board to the other side of the board that isn't intersected by one domino.
In the trivial case you have exactly $3$ dominoes lined up in each column or row of the square. Then you'll get $5$ lines verticaly and $2$ lines horizontaly and you're done.
Please give me only a hint on how to proceed proving this generally.
This isn't an answer, but an observation too long for a comment.
Note that an $8 \times 8$ board can be covered with dominoes in such a way that every line is blocked:
The fact that this is possible means that an induction argument isn't the right way to go; with induction, the point is typically that you can continue a pattern ad infinitum. In this case, you need to make an argument specific to the $6 \times 6$ board (as Jaap Scherphuis pointed out in the comments).