An instant-win lottery game system includes a plurality of instant-win lottery tickets, with each of the tickets having a play area with a plurality of play spaces arranged in defined pattern. Randomly generated play symbols are depicted in a plurality of the play spaces that is less than all of the play spaces. A plurality of line segments are also depicted in the play area, with each line segment connecting at least two play spaces. The line segments have a placement and orientation in the play area that is randomly generated. A plurality of the play spaces that is less than all of the play spaces are connected to at least one other play space by the randomly generated line segments such that different lottery tickets in the game system have the same number of connected play spaces, but with different patterns of randomly generated line segments. A scratch-off layer is provided over at least a portion of the play area and covers the randomly placed line segments and play symbols. Markings on provided on the scratch-off layer that provide at least a partial indication to the player of the placement and orientation of the underlying line segments.