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## An Endgame

King and two Rooks versus King.

#### First Phase

White Rooks on 3c3 and 3c6. At this distance, their moves complement each other, and they cover all but the corners of a 5x6 plane on the c-file. (Remember that their moves aren't symmetrical, so they can cover a wide area on a vertical plane, but not on a horizontal plane).

Imagine the Black King on one of the lower levels of the left side of the board. In order to cross the c-file, the black K must get to 1c2, 5c2, 1c7, 5c7, or some other squares that are farther away from the Rooks. When the King approaches these squares, the Rooks can shift their blockade with a move such as R3c3-6c6; and in the meantime, the White King has free tempi, so he comes in and helps.

It seems reasonable to assume that the Black King can eventually be confined to a corner.

#### Second Phase

White King 1b3, White Rook 2d1, Black King 1b1. Black is confined and not stalemated, and it is easy to mate with the other Rook.

## A Mating Position

WHite Pe2, Qe5, Ke6; Black Ke4; all on the same level. Once again, this is also mate in the 2D game.

## The Shortest Game Possible

1. 4e2-4e4 4e7-4e5 2. Q4d1-4h5 K4e8-4e7 3. Q4h5:4e5 mate

The normal foolsmate does not work in 3D, but this pretty mate can be accomplished in 3D using seven different White Queens, one of which has 3 routes. (You might want to check and see if I'm wrong.)

## Pawn Power In Chess

In the normal games of Chess, each square of the board can potentially be attacked by at most two Pawns, so that every time you advance a Pawn you need to think about the squares for which you have weakened your potential for Pawn control.

In 3D Chess, each square is potentially attacked by as many as 8 Pawns, and therefore Pawn weaknesses are not as pressing. After, for example, 1. 4e4 N4g8-4f6 2. 4e4-4e5 N4f6-4d7 3. 4e5-4e6? Black can capture any of 6 ways, and though the Pawn at 4e7 is a bit blocked there are other e-Pawns that can be moved to get the pieces developed, and of course Black still can attack 4e5 with a Pawn -- in fact, with any of 6 Pawns.

However, Pawns still have very limited mobility, which was designed so that Pawn chains can still be formed and play their normal part in setting up walls that make the strategic terrain more interesting. It may be a problem that the weaknesses on both sides normally caused by a Pawn chain will be absent.

For example, after 1. 4e4 4e6 2. 4d4 4d5 3. 4e5 4c5 4. 4c3 4c4, we have a perfect 2D Pawn chain. Moving into the third dimension with 5. 5d4, we confront the question of what a 3D chain should look like. (Of course, Black's 4th move here is probably wrong.)

Notice that 6. 5d5 is now legal, attacked by 4e6 and defended by 4d4. In addition, 5. 5d4 invites the reply 4c4:5d4 en passant, which is presumed bad because it dissolves Black's leading Pawn.

Therefore, 5. 5d4 5d5 6. 3d4 3d5 7. 4d4:5d5!? Q5d8:5d5

Black couldn't recapture with 7...4e6:5d5 because then 4d5 would be attacked 3 times and defended only once. The shape of the two chains right now is interesting. White's 7th move is probably wrong.

8. N6b1-5c3 attacks 4d5 4 times, and also attacks 5d5. There's no pin. Too bad that 8...4c4:5c3 is legal!

8. N7b1-6c3 can be pinned (B2f8-7b4, Q4d8-8a5), but neither move is legal. A possible reply is Q5d5-4e4+, which looks like a bad move. After the Queen moves away, the forcing way to play would be to take 3 times on 4d5 but then 4e5 would need defending. Might as well play this and pick a random move for Black's Queen (but not 8d5 of course).

8. N7b1-6c3 Q5d5-4e4+ 9. Q3d1-4e2 Q4e4:4e2+ 10. C3e1:4e2

As it turned out, I didn't like the other moves either. Remember that if 8...Q5d5-5a5, 4a2-4a4 attacks the Queen and opens up moves like R3a1-6a4 (so it helps White's development). Maybe I just didn't see the right move, but I decided Black should aim to end the forcing sequence with sente, even at the cost of a minor development tempo for White.

10. ... 5e7-5e6 11. 5e2-5e4 3e7-3e6 Black thickens the chain, White reinforces and then some. 5e4 defends 4e5 and is defended by the N6c3; it also ignores 3d5 and attacks 4d5; and 4d5 is now attacked by 3d4, 5d4, 5e4, and Q4d1 (and 4d5 is defended by 4e6, 5e6, Q4d8). Black is happy to defend, because in the meanwhile he is attacking 4e5.

In fact, since 3e4 is attacked by two Black Pawns, White has no more center Pawns that can defend 4e5. If White pushes all of 3f4, 4f4, 5f4, Black can simply go 3f6 and so on. A capture on 4d5 would help relieve the pressure on 4e4, but helps unblock Black's position.

12. 3f2-3f4 3f7-3f6 13. 4f2-4f4 4f7-4f6 14. 5f2-5f4 5f7-5f6 15. N2g1-3f3

White simply aims to keep that square. After all the Pawn captures that could start here, Black would have doubled isolated e-Pawns in 3 dimensions, and at the end White plays N3f3:4e5.

1. 4e4 4e6 2. 4d4 4d5 3. 4e5 4c5 4. 4c3 4c4 5. 5d4 5d5 6. 3d4 3d5 7. 4d4:5d5!? Q5d8:5d5 8. N7b1-6c3 Q5d5-4e4+ 9. Q3d1-4e2 Q4e4:4e2+ 10. C3e1:4e2 5e7-5e6 11. 5e2-5e4 3e7-3e6 12. 3f2-3f4 3f7-3f6 13. 4f2-4f4 4f7-4f6 14. 5f2-5f4 5f7-5f6 15. N2g1-3f3 6d7-6d5 16. 5e4:4d5 5e6:4d5 17. 3d4:4d5 3e6:4d5 18. 5d4:4d5 4e6:4d5 19. N4g1-4f3 B2f8-7b4?! 20. 7a2-7a3!? B7b4:N6c3 21. N6b1:N6c3 N4b8-4c6 22. N5g1-5f3 N5b8-5c6 23.

I can't think about this position any more, but have seen enough to conclude that 3D Pawn chains do not form an immobile plane of locked Pawns (their extensive capturing power prevents this), but are still rather important in all dimensions.

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