MathDB

m,n,p

are positive integers which do not simultaneously equal to zero.

3

D Cartesian space is divided into unit cubes by planes each perpendicular to one of

3

axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from

1

60

. A filling of integers is called Dien Bien if, for each rectangular box of size

\{2m+1,2n+1,2p+1\}

, the number in the unit cube which has common centre with the rectangular box is the average of the

8

numbers of the

8

unit cubes at the

8

corners of that rectangular box. How many Dien Bien fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation.
translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here

Problem Statement