For example, before the first move the alternate board is empty, but after 1. e2-e4, there is a single White Pawn on the alternate board. After 1. e4 e5 2. Qh5 Nc6 3. Qh5:e5, the White Queen comes back to the original board and it is checkmate.
In the context of Chess Variants with Different Armies, we have to wonder if Alice's Chess is stronger than FIDE Chess.
The Alician Army has a whole extra board to play with, a board where its pieces cannot be threatened or blocked by the FIDE pieces; but on the other hand, when the Alician pieces are on the extra board, they can neither threaten nor block the FIDE pieces.
However, it is not certain that this rule is necessary! Certainly it would be a wilder variant to play flat-out Alice on one side, with Alician Pawns and an Alician King.
Later on, I give a Sample Game with Alician King, which seems to demonstrate that the Alician King is too strong.
In Alice's Chess, you could reach this Pawn formation with 1. e3 e5 2. e4 Nc6 and 3. e4-e5 is not a legal move because the Pawn would first arrive on e5 of the original board and then it would need to step through the looking glass onto the square e5 of the alternate board -- but the destination is occupied and so the move is illegal!
However, in our example, the Pawn that goes to e5 does not need to go onto the other board, and so perhaps there is no reason for the move to be illegal.
Surprise! The White Pawn at e5 would be immune from capture! This would be unfair, and so it is illegal for White to play a piece to any square that is occupied on the other board.
Therefore, the rule must be that Alician Pawns are immune to en passant capture, but FIDE Pawns are vulnerable!
This rule, like the whole game, is deliciously complex and arcane.
White plays FIDE, Black plays Alician, both use the FIDE army.
1. e4 e5 2. Qh5 Qh4!
Maybe not a good move, but a surprising one.
3. Bc4 Nf6
Note that 3...Qe7 4. B:f7+ Q:f7 is illegal because the Queen disappears onto the other board and leaves the King in check.
4. Q:f7+ Kd8 5. Nf3 Qg4!
Threatens to take the Knight and run away to safety on the other board.
6. Nd4!? Q:e4 7. Nb5 Nc6 8. N:c7 Rb8 9. O-O Bc5!? 10. Q:g7 Rg8 11. Q:h7 Rb4
Black could not keep all his Pawns, but now plans to use the extra board to develop a smashing attack. White cannot defend any pieces against the hit-and-run attack.
12. Bd5 Rg5 13. d4 R:g2 14. Nc3 Rg2-g4+ 15. Kh1 Rb4:d4
The threat is Rd4-h4, Rh4:h2, and Rh2-h3 mate. Although there are defenses, the position feels to me like Black is winning, and mainly because the Black King is so safe that there is no counterplay.
Threat 6. Bc1-g5 mate.
5. d4 Qg4 6. h3!
In the second game, quite possibly my choice of Black's moves was awful and deserved to lose fast. Anyway, it did lose fast.
However, it certainly looks like having an Alician King with an Alician army is very strong, and also that having a normal King with an Alician army makes it hard to defend yourself.
The natural thing to try next would be to have a non-Alician King and Pawns, several non-Alician pieces for defense, and a few Alician pieces for offense; but the Alician pieces should not be *too* strong!
The result of this should be a game where both sides race to see who can give mate the faster.