MathDB

Geodude wants to assign one of the integers

1,2,3,\ldots,11

to each lattice point

(x,y,z)

in a 3D Cartesian coordinate system. In how many ways can Geodude do this if for every lattice parallelogram

ABCD

, the positive difference between the sum of the numbers assigned to

A

and

C

and the sum of the numbers assigned to

B

and

D

must be a multiple of

11

? (A lattice point is a point with all integer coordinates. A lattice parallelogram is a parallelogram with all four vertices lying on lattice points. Here, we say four not necessarily distinct points

A,B,C,D

form a parallelogram

ABCD

if and only if the midpoint of segment

AC

coincides with the midpoint of segment

BD

.) [hide="Clarifications"]
[*] The ``positive difference'' between two real numbers

x

and

y

is the quantity

|x-y|

. Note that this may be zero. [*] The last sentence was added to remove confusion about ``degenerate parallelograms.''
Victor Wang

Problem Statement