From 3D to 4D in four steps
The goal is still checkmate. White moves first. Every rule below applies equally to all four axes, and that single idea is the whole game.
1. Give your 3D board one more coordinate
Think of a 3D chessboard as layers of ordinary boards. X and Y locate a square within a layer; Z chooses the layer. Here each 3D board has two 4 × 4 layers: the left board is Z = 0 and the right board is Z = 1.
Now make a second copy of that whole 3D board. W chooses which copy you are in: the bottom row of boards is W = 0, the top row is W = 1. Every square has a twin in the other copy, and a move along W changes copies while keeping X, Y and Z the same.
In the tesseract view, W = 0 is the inner cube and W = 1 is the outer cube. Both are the same size; the inner cube only looks smaller because of the projection. W is a spatial direction, not time.
| Axis | Where it takes you |
|---|---|
| X | Across a board, files a to d. |
| Y | Up a board, ranks 1 to 4. |
| Z | Between the left and right boards in the same row. |
| W | Between the bottom and top boards in the same column. |
a1[0,1] means file a, rank 1, Z = 0, W = 1: a1 on the top-left board.
2. See every piece's reach
Pick a piece. It sits on b2 of the bottom-left board with the rest of the board cleared, and every square it can reach lights up on the flat boards and in the tesseract.
3. Count the coordinates that change
Judge a move by its coordinates, not by how it looks. A W move can look diagonal in the drawing and still be a straight rook move. Rotating the view changes how a move looks, never whether it is legal.
Change exactly one coordinate by any distance.
Changing W and Z together is a diagonal, so a rook cannot do it.Change exactly two coordinates by equal distances.
X + W or Z + W works. X + Y + W changes three axes and does not.Change one coordinate by two and a different coordinate by one. Jumps over pieces.
The two-step part must use X or Y; Z and W only have two positions.Change any non-empty combination of coordinates by equal distances.
One step on all four axes is legal. Two on X and one on W is not.Change any non-empty combination of coordinates by one step.
Every destination must be safe, including from threats in the other cube.Advance one empty square along Y. Capture one Y step forward plus one step along exactly one of X, Z or W.
White advances toward rank 4, Black toward rank 1. Reaching it makes a queen.Any bishop or queen move involving Z or W is at most one step long, because those axes only run from 0 to 1. X and Y each have four positions.
4. Look for threats across W
A king can be in check from a piece in the other cube. A rook on a1 of the bottom-left board attacks a king on a1 of the top-left board along W. Distance on the screen gives you no protection.
You may capture the attacker, block a longer sliding attack, or move the king, as long as the result is safe. A one-step W attack has no square in between to block. Two lines crossing in the projection do not necessarily share a square: a piece blocks a sliding move only when its full X, Y, Z, W position lies on the path. Knights jump.
Setup, special moves and draws
The full set gives each side the complete chess army: king, queen, two rooks, two bishops, two knights and eight pawns. White's back rank runs R N B Q on the left board and K B N R on the right board of the bottom row, pawns on rank 2 of both; Black mirrors it on the top row. The classic set from the original page has ten a side: king, queen, two rooks, a bishop, a knight and four pawns. Either way there is no castling, en passant or opening double pawn move.
The game is drawn by stalemate, the third occurrence of the same position with the same side to move, 100 moves without a pawn move or capture, or two bare kings. The king is never captured. Rated games use Glicko-2, the same rating system as chess.com and lichess.
Keyboard: Tab to a square, Enter or Space to select or move, arrow keys within a board, Alt + arrows between boards, Escape to clear.
Try it
You are White. Try the rook on a1 to the board above: a1[0,0] → a1[0,1]. A small computer answers.
These rules follow the open 4D chess variant on a 4×4×2×2 board, where the same movement rule applies to every axis.