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This game does look like it needs a ref to be able to ref things. And the game should be like chess in every regarding, including the need to move the King if the King is in check. It makes the rules easier on people.
Of course, the players do not have to inform the other that their move resulted in a check. Only that they've moved their King to remove this possible check. The attacking player could then suppose that the previous position resulted in a check. And simply making a King move could fool an opponent into thinking that they had made a checking move. And thus draw them into a negative position. Also, players could play on-line. They would send the TXT of each King move in a password-protected ZIP file. At the end of the game, they would then exchange passwords. Allowing each to verify the legality of the game played. And, this could also work in the real-world. Each King move and its turn number is written on a slip of paper and dropped into a container. Which is then checked at the end of the game. This would still mean that a record of play needs to be kept for the entire game.
It appears the Black brothers were aware of Betza's Lag Chess by similarity and mention of Kriegspiel near the end, as my comment back 10 implies. Not to say it is not worthwhile spinoff by CVPage standards.
The only stumbling block to a totally referee-less game is the potential of King threatening King. But what if the Invisible Kings were allowed the privilege of not attacking one another and could even occupy the same cell?
You are right that is another way. Have it the same as chess with regard to check (so you cannot move into check or stay in check), and no notifications about check. It is variant of the variant.
You can play this game using the following ZRF (which I created in 2003): http://www.zillions-of-games.com/cgi-bin/zilligames/submissions.cgi/47043?do=show;id=554 Certainly, Zillions-of-Games will see where your king (Zillions doesn't really support games with hidden information). The computer's king is hidden from human player by making its image transparent (so it is invisible).
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