The number of paths of length 4*n* from one corner of an
*n* × *n* × *n* × *n*
lattice to the opposite corner is

;

the first few values (starting with *n* = 0) are
1; 24; 2,520; 369,600; 63,063,000; 11,732,745,024; 2,308,743,493,056;
472,518,347,558,400; 99,561,092,450,391,000; 21,452,752,266,265,320,000;
4,705,360,871,073,570,227,520; ….
(Compare this to the 2-D and
3-D versions of the same idea.)

There are 24 possible paths of length 4 through a 1 × 1 × 1 × 1 lattice from one corner (0,0,0,0) to the opposite corner (1,1,1,1).

For a 2 × 2 × 2 × 2 lattice*, there are 2,520 paths of length 8 from the green dot (0,0,0,0) to the red dot (2,2,2,2).

Here are the first eight:

Here is a “middle” path, from (0,0,0,0) to (0,0,0,1) to (1,0,0,1) to (1,1,0,1) to (1,1,0,2) to (2,1,0,2) to (2,2,0,2) to (2,2,1,2) to (2,2,2,2):

And here are the last eight:

^{*} You’ve surely noticed that the positive
*x*, *y*, and *z* axes point in the usual directions,
and that the positive *w*-axis points “inward”; thus
*w* = 0 corresponds to the outer cube,
*w* = 1 the intermediate cube, and *w* = 2 the inner cube.
The trouble is, I had no idea what to do with the points (1, 1, 1, *n*), so
I did nothing: they all map to the unattached point at the center of the whole arrangement.
If you have a better idea, please let me know.

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