OFFSET
3,1
COMMENTS
Developed as the three-dimensional extension of the Mondrian Art Problem.
Alternatively, a(n) is the minimum defect when an n X n X n cube is partitioning into four cuboids of different dimensions.
Let elements of the unordered integer triplet (x,y,z) be the dimensions of a cuboid in a set S of four cuboids.
Let V(x,y,z) = x*y*z be the volume and for a given element of S. Define min(S) = min{V(x,y,z): (x,y,z) in S}, max(S) = max{V(x,y,z): (x,y,z) in S}, and defect(S) = max(S)-min(S).
a(n) is the smallest value of the defect(S) across all possible partitions of the n X n X n cuboid into four cuboids of different dimensions.
LINKS
Janaka Rodrigo, Python program
EXAMPLE
The 4 X 4 X 4 cube has 12 different ways of partitioning into four distinct cuboids each giving a defect as the difference between the largest volume and the smallest volume relevant to the four cuboids of the set.
The optimal solution is given by the set {(4,2,1), (4,2,2), (4,3,2), (4,4,1)} because it has the minimum defect.
The least possible defect = max(8,16,24,16)-min(8,16,24,16) = 24-8 = 16.
Therefore, a(4) = 16.
CROSSREFS
KEYWORD
nonn
AUTHOR
Janaka Rodrigo, Jun 19 2025
EXTENSIONS
More terms from Sean A. Irvine, Jul 16 2025
STATUS
approved
