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Proysen: Ohhhh! I didn't grok the part about the markings on the cards serving as blockers. In that case, your variation isn't nearly as broken as I thought.
White's 2468 piece is a very safe play, however, since no line can go through it.
The illustration I provided is biased towards white for demonstration purposes only. In a real game, both sides would have an equal amount of pieces in play.
Proysen, I'm 99.9999% certain that rotating a piece is allowed. The acrylic edition of the game doesn't feature logos on the pieces to establish any one orientation as more "correct" than the other ones.
Also, your proposed variant seems to break the game. If White opens with the 2468 piece, he cannot possibly be blocked from making a winning arrangement in the next 4 turns. If Black blocks the space above, White just plays the space above that, and similarly for the other 4 directions.
A very fun thinker's game nonetheless.
Also glad to see someone else plays and enjoys Matthew Brown's games, you have great taste.