Let the coordinates of (infinite-dimensional) Hilbert space H
be denoted as x_1, x_2, x_3, ..., x_n, ....
Note that the integer points of H correspond to arbitrary sequences
of integers *only finitely many of which* are non-zero.
The cubic tiling of H has one cube for each integer point of H.
Definition:
-----------
The cube at the integer point x is "adjacent" to the cube at
the integer point y exactly if y differs from x in precisely one
coordinate, by the amount ±1 in that coordinate, and has all
the other coordinates equal to those of x.
Puzzle:
-------
Prove that given infinitely many colors, indexed by the set of
integers, the cubes of the cubic tiling of H can be colored
so that each cube is adjacent to exactly one cube of each
other color (and not adjacent to any cube of its own color).
—Dan
