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Let's say that on an 8x8x8 board rooks can move any number of spaces, only moving through one dimension at a time. Bishops move the same number of spaces in exactly 2 dimensions at a time. Knights move 2 spaces in 1 dimension and 1 space in another dimension. Kings move 1 square in any direction, meaning that in the center of the board a king controls 26 spaces.

I noticed that in 2 dimensions rooks, and bishops have the same number of directions they can move in, but in 3 dimensions with the above way to generalize the movement of the pieces to 3 dimensions, there are twice as many directions to move for the bishops as there are for the rooks. Also the ratio between the number of spaces bishops control to the number of spaces knights control is the same in 2 dimensions and 3 dimensions. A King and Rook is not enough to checkmate a king in 3 dimensions as the lone king can always use the 3rd dimension to escape any check from the rook. In 2 dimensions a knight takes 4 moves to get to the spaces that are 2 spaces diagonally away, but in 3 dimensions because of the extra degree of freedom a knight only takes 2 moves to get to the spaces 2 spaces diagonally away from it. This makes it seem like Bishops and Knights might be more powerful than rooks, and that knights might be more powerful than bishops as they don't have all the same blindspots they do in 2 dimensions.

So how powerful are Rooks, Bishops, and Knights on an 8x8x8 board compared to each other?

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    It's worth noting that bishops are still colour-bound, which is a definite weakness compared to a rook. Dec 18, 2022 at 23:51

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I would say yes, the bishops and knights would probably be slightly stronger than the rooks.

Let's start out with a look at how many squares the pieces cover on an empty board:

Traditional chess:

  • 64 squares on board
  • Rook: 14 squares
  • Bishop: 7-13 squares
  • Knight: 2-8 squares
  • King: 3-8 squares

Variant:

  • 512 squares on board
  • Rook: 21 squares
  • Bishop: 21-39 squares
  • Knight: 6-24 squares
  • King: 6-26 squares

If we were to simply multiply based on square control, if knight-bishop-rook used to be 3 - 3.25 - 5, it would now be 9 - 9.75 - 7.5. If we were to then divide by 3 to take into account the increased power of the king, it would be 3 - 3.25 - 2.5. So by this metric the rook would indeed be weaker.

The knight's "blind spots" would be reduced. It would still, however, take 3 turns to get to an adjacent square. A knight in the corner could reach any square in 8 moves instead of 6 due to the bigger board; the loss of that blind spot might be offset by the increased number of squares not close to the knight. And a knight would still be unable to gain or lose a tempo.

You don't mention the pawn rules; they would probably have an impact on how easily a knight outpost could be created. Once created, it seems like it would be more difficult to keep an enemy knight out. A well-placed knight with 24 possible squares it could go to would seem fairly dangerous; a knight on the 4th level could threaten at least some squares in the 2nd through 6th levels directly, and could threaten forks in some places in all the levels.

The bishop would be much more difficult to trap or restrict. A bishop isn't very "bad" if it can just go over or under the pawns, after all. Visualizing their new movement might take some getting used to. They would still be restricted to one color, of course, which does limit their power just like it does in regular chess.

A bishop can't really dominate a knight on the side of the board like it can in 2d chess. The bishop can prevent the knight from going in a particular direction for one turn, but cannot maintain this.

A pair of bishops would no longer be able to keep an enemy king on the side of the board by themselves. Bishops have always, of course, had a blind spot directly in front of them, but that could be protected by the other bishop in the pair. But now they gain additional blind spots in the squares which are diagonal in all 3 dimensions. There is a 2 bishop + king helpmate available at the corner of the board, but I'm guessing it can't be forced.

Bishops would gain the ability to break a pin while still protecting the pinned piece. If, for example, a bishop was on c1 protecting a pawn on d2 from an enemy bishop on h6, the d2 pawn would be pinned - but the c1 bishop could move to d1 on an adjacent level, which still protects the pawn but breaks the pin. Rooks would not gain such an ability; a rook on d1 with a pawn on d2 and an enemy rook on d8 does not have a square it can get to in one move where it could break the pin while still protecting the pawn. (Kings and queens already have this ability in 2d chess, and it's impossible for that situation to come up with a knight in either game; there's no way for a knight to be behind a pin while directly protecting the pinned piece.)

Rooks would lose their property of being able to reach any square in two moves on an open board; now it would take three. And, of course, a rook can no longer give checkmate with only the king to help. Back rank checkmate threats would also seem to be reduced if the enemy king can escape via the other dimension. But they are still a long-range piece which can perform pins and forks.

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  • 8x8x8. Is this 3d chess now?
    – ShadYantra
    Dec 19, 2022 at 8:17
  • Yep, question mentions 3 dimensions.
    – D M
    Dec 19, 2022 at 21:44

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