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The number of paths of length 4nfrom one corner of annxnxnxnlattice to the opposite corner is;

the first few values (starting withn= 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 x 1 x 1 x 1 lattice from one corner (0,0,0,0) to the opposite corner (1,1,1,1).

For a 2 x 2 x 2 x 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 positivex,y, andzaxes point in the usual directions, and that the positivew-axis points "inward"; thusw= 0 corresponds to the outer cube,w= 1 the intermediate cube, andw= 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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