With what I have seen in the anime, this should sum up all the rules and factors.
- The objective is to capture the king.
- The game is played on a single color 9x9 board.
- The game starts with a empty board
- The players take turns placing stones, limited to the first 3 rows of the board on your side of the field
- After all the pieces are placed, the players take turns moving a piece.
- The pieces can be stacked on top of each other (the 3rd dimension) up to 3 pieces.
- Certain pieces are stronger than others. Therefore, not all setups can beat certain counters.
- Each player has 24 pieces in total.
The pieces used in this game are as following
The exact conditions for victory are too blurry to confirm, but the rules for victory seem pretty similar to chess, which would equal to trying to checkmate the king. The placement and tactics used in the game are pretty similar to shogi. As for the initial placement, it's pretty similar to checkers. For the combat side of the game, it also seems to have a slide of Stratego in it, as some pieces are stronger than others, and there potentially are pieces that can only be beaten by certain other pieces.
This is all I managed to find out.