I have been thinking about writing a program that fills a 4x4x4 cubical
box of cells, each cell with a single digit (0 to 9), such that all of
the integers can be found by stepping from one cell to a neighboring
cell. Neighboring cells share a face, edge, or vertex. One way to do
this would be by ensuring that each occupied cell has neighboring cells
with every digit 0 to 9. Since at least 10 neighboring cells would be required to handle all digit sequences, the eight corner-cells would be
of little use, since they only have seven neighbors. My plan was to
leave corner-cells unoccupied. Edge-cells are part of a 2x2x3 sub-box,
so there would be just enough neighbors (11, or only 10 if the sub-box
has an unoccupied corner-cell within it). Mid-face and interior cells
have even more neighbors.
Putting every integer into a such a small "Universal Integer Box" would
be gratifying.
Is this "Universal Integer Box" even possible?
What's an efficient algorithm for filling the box,
Can a smaller box work (e.g., 3x4x4)?
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