3 Dimensional Cubic Chess 8x8x8 Board

2-2-1 leaper is different enough I don't think it qualifies as a knight movement anymore. You're talking [radical 9] now as the length of move instead of 5 or 6, that is a huge difference and wouldn't consider those the closest cubes. Same for any 3-2-1 piece, even though that concept is cool!

Yes that was what I liked the best. The 48 comes from the 24 [2-1] moves and the 24 distinct [2-1-1] moves as well. Axis is the best word to explain it, thanks, the "1" part of the move can be diagonally away from the "2" part since it is still perpendicular. There is one ugly implication of this though. This would let a knight fork 2 pieces that are right next to each other, which is really weird. A standard 2-1 knight simply utilizing all 3 dimensions with 24 possible moves would maintain the relative power ratio of the pieces, as in 2d chess, if both pieces are in the center, 8 knight moves 8 king moves. In 3d chess, 26 king moves and 24 knight moves. So 48 ends up seeming weird in that respect, like the knight is now almost twice as powerful as the king. Maybe an alternative would be to break it up between 2-1 and 2-1-1 leaders as separate pieces, but have nothing that could do both. So hard to decide lol

Oh and actually it is 48 when you combine the two, not 72, as the 2-1 moves shifted diagonally are redundant to the 2-1-1 moves.

Here's a helpful visualization:

3d Board level 5:

3d Board level 7:

These are the pawns that could be taken via 2-1 moves. Taking into account 6 directions = 24 total 2-1 moves.

3d Board plane 7 again:

These are the pawns that could be taken via 2-1-1 moves. Again the number is 24 for the same reason, 4 cubes X 6 directions for the initial "2" move.

So the total is 48 for both, 24 for each. Not 72.

Right answer, but I think the reasoning is different.

Each sequence of 3 distinct signed numbers gives a distinct location. So there is no overlap between 2-1-0 class and 2-1-1 class.

2-1-0 gives 6 choices of axes, but only 4 choices of signs. So 24 locations. (I included a sign for 0 to wrongly get 48).

2-1-1 gives 3 choices of axes and 8 choices of signs. So another (non-overlapping) 24 locations.

A knight with only a 2-1-1 move pattern would be restricted to half, possibly even a quarter of the cubic board. Not sure how to calculate which though.

Another reason why 2-1-0 is best.

The answer to your question is that it's half. To see this note that (2,1,1) + (-1,-1, -2) = (1, 0, -1). So you get all the 1-1-0 moves, which covers half the cubes. Parity (evenness of sum of moves in each of three directions) means you can't get more than half.

3d bishops are restricted quarterly though I think. (1,1,1 or 5,5,5..etc triagonal moves)

So your 3d bishop only moves N-N-N?

That gives you 2-0-0 which gives you 1/4 of cubes, but that is all.

To prove that informally, the parity (even/odd) of each dimension of a reachable location always has to be the same as the other two dimensions (because it is for each basic move). That means that you can't get to 1-0-0 or 1-1-0 or 2-1-0 (and same with signs) which leaves you with exactly 1/4 of the locations on the x-x-0 plane, and the same for any other plane, since you can get to at least one location in each.

No, that's why I don't prefer only 3d diagonal moves for the bishop, and my 3d bishop moves both like a 2d bishop and a 3d bishop, just like the knight. It basically can move to any cube a kings move away except the rook-like moves away. 60 cubes total I calculated. A bishop on "5d5" for example:

5th level:

Could move to these cubes:

Level 5:

Level 4:

Level 6:

Level 2:

Level 8:

Get the idea?

It's also interesting to note that bishops that move only triagonally is restricted quarterly, can only access 1/4 of the cubes (think of the four triagonal moves possible within any 2x2x2 sub-section). An exclusively 3d bishop on any one of those lines could only move back and forth between the 2 cubes on that line. 2 cubes on different triagonal systems could never both be reached by a 3d bishop, which is why I included 2d moves.

There is so much interesting information here.

Not to brag but probably the only actually interesting chess "variant". Although it's a dimensional extension, not a variant. No changes are being made to the rules, board size, or piece types. Everything is bring scaled up 1 dimension.

You need 8 boards irl

Would have to be a digital interface really.

Movement is that it can turn upwards or downwards

You can play it with less 3d players, i ma do it irl

I did it and it was a lot of fun, the movements of pieces turning upwards

Knights are very powerfull

Queens are devastating