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How do you figure Black can offer check on his first turn? He can only get a pawn to rank 4 on his turn, which only threatens ranks 2 and 3. What is missing?
With weak players, the double moves are too hard to anticipate. For stronger players, after each loss for the smaller army, add a pawn. After each win, subtract a pawn until things get competitive.
Although advancing the King and Queen pawns ought to work, as you suggest, what about a double flank attack? To do so, try fianchettoing both Bishops, and then follow up by bringing the Knights out to the Rook files (A3 and H3), and then move them up for sacking on Black's pawns? In response to that kind of a strategem, it appears that throwing the Knights and Bishops away is a necessity, not a convenience. And if White tried to do that, wouldn't Black's best plan be to refrain from going forward, and instead sit around moving his King back and forth? Can either side 'pass' or is moving mandatory?
I played the game many times. If White plays defensively, moving the two middle pawns out only 1 space, and backing them up with the bishops and if need be the queen, black won't win.
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Should read: "After the first move, the Black king may be in check"
There is a reference to the game in The Machine Plays Chess?