The (solid) tesseract can be described as the set of points (x,y,z,w) with every coordinate between 0 and 1. The 2d faces then are characterized by setting exactly two coordinates to either 0 or 1 (and letting the other two coordinates range between 0 and 1).
Subdividing those faces into a 5x5 checkerboard, the centers of the squares are points with exactly two coordinates equal to 0 or 1 and the other two among {0.1, 0.3, 0.5, 0.7, 0.9}. Perhaps then a usable coordinate system just replaces those decimals by a,b,c,d,e respectively.
In that system, one rook path would start
01cc, 01dc, 01ec,
at which point the rook hits the edge point "011c" (not a playable square) and the two continuation paths must move off the extreme value of either of the first two coordinates:
, a11c, b11c, c11c, ...
, 0e1c, 0d1c, 0c1c, ...
The (solid) tesseract can be described as the set of points (x,y,z,w) with every coordinate between 0 and 1. The 2d faces then are characterized by setting exactly two coordinates to either 0 or 1 (and letting the other two coordinates range between 0 and 1).
Subdividing those faces into a 5x5 checkerboard, the centers of the squares are points with exactly two coordinates equal to 0 or 1 and the other two among {0.1, 0.3, 0.5, 0.7, 0.9}. Perhaps then a usable coordinate system just replaces those decimals by a,b,c,d,e respectively.
In that system, one rook path would start
01cc, 01dc, 01ec,
at which point the rook hits the edge point "011c" (not a playable square) and the two continuation paths must move off the extreme value of either of the first two coordinates:
, a11c, b11c, c11c, ...
, 0e1c, 0d1c, 0c1c, ...