This seems similar (possibly equivalent) to problem B6 on the 2019 Putnam. https://kskedlaya.org/putnam-archive/ On Wed, Sep 30, 2020 at 7:14 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:
Centre d-cube at x in |Z^d , where 0 <= d <= |N ;
Denote colour C by integer in {-d, ..., +d}
Define C(X) = \sum_{0 <= i <= d} i x_i mod (2 d + 1) ;
the sum remains finite since bounded by |x| max(i | X_i <> 0) .
WFL
On 9/29/20, Dan Asimov <dasimov@earthlink.net> wrote:
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
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