The number of possible paths of length 5*n*
from one corner of an *n*-by-*n*-by-*n*-by-*n*-by-*n*
lattice to the opposite corner can be calculated using this formula:

The first few terms are 1, 120, 113400, 168168000, 305540235000, 623360743125120, ... (Compare this to the 2-D, 3-D, and 4-D versions of the same idea.)

The empty 1-by-1-by-1-by-1-by-1 lattice looks like this, where each pair of “adjacent” points is joined by a line, and the starting and ending points are highlighted:

Each step of the path can occur in one of five directions.

For the 1-by-1-by-1-by-1-by-1 case, we’re counting paths of length 5; since each path
will be made up of one step in each of the five directions, we can easily enumerate the paths
by computing all the permutations of (*dir*_{1}, *dir*_{2}, *dir*_{3}, *dir*_{4}, *dir*_{5}).

For the 1 × 1 × 1 × 1 × 1 lattice, here are the 120 (that is, 5!) paths:

Designed and rendered, at one time or another, using *Mathematica* for the Apple Macintosh, Microsoft Windows, and for NeXT.

© 1996–2023 Robert Dickau.

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