## Some Sample Endgames

The positions discussed here are valid both for 3D Chess 1.0 and 3D Chess 1.1 except where so noted.

In general, at the end of the game you will need more than a King and Rook to checkmate a bare enemy King. King and Queen versus King appears to be a draw.

However, in 2D Chess, a Rook is about one-eighth of your entire starting force. In 3D Chess, you need considerably less than one-eighth of your starting force to checkmate the bare enemy king.

Imagine your King at c1, and the enemy Q at c2, defended. It's checkmate, 3D or not. With Queen plus King versus a bare King, it is possible to mate but not (I think) to force mate.

In fact, if all of the pieces in a diagram are on the same level, a 2D checkmate is often a 3D checkmate.

### King and Queen versus King

Start with the White King at 3e3, the Black King at 1e3. White to play, a White Queen is somewhere.

If White can force checkmate from this position, there is hope that White can, in the general case, force the King to the side and win; but if White cannot even win from this favorable starting point, it is certain that this endgame, in the general case, is a draw.

Clearly, if White can play Q-2e3, the game is over. If this is not so, White must be able to deal with 8 different Black King moves. Bringing the Queen to 1c4 prevents many of them.

Now we have WK 3e3, WQ 1c4, BK 1e3, Black to play, and the obvious 1...K1e3-1f3 threatens to escape with 2...K1f3-2f4 and a few other places.

There is no way to confine the Black King, so instead we should try 1. Q-2d4+; now because many moves are checkmate, for example 1...K1e3-1f4 2.Q2d4-2f4, Black must go to 1f3 or 1e2.

After 1. Q-2d4+ K1e3-1f3 2. K3e3-3f3 K1f3-1e2!, Black threatens to escape via 1d2, and the game is a draw.

I believe that K+Q cannot win without extra help; if the extra help is another Queen or Rook, of course White wins.

### King, Queen, and Commoner versus King

With White K 3e3, White Commoner 4e3, Black King 1e3, White Queen 2d4+, we have 1...K1e3-1f3 2. (WF)4e3-3f3 K1f3-1g3 2. Q2d4-2f4+ K1g3-1h4 3. (WF)3f3-2g4 mate.

This is a very favorable starting position, and I haven't analyzed it completely. Quite possibly it is a draw in the general case, but I think it might be a win.

### King and Two Rooks verus King

Here is an analysis of the mate with 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.

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