cp's OEIS Frontend

A129668 Number of different ways to divide an n X n X n cube into subcubes, considering only the list of parts.

%I A129668 #48 Feb 16 2025 08:33:05
%S A129668 1,2,3,11,19,121,291,1656
%N A129668 Number of different ways to divide an n X n X n cube into subcubes, considering only the list of parts.
%C A129668 The Hadwiger problem analyzes how to divide a cube into n subcubes. This sequence analyzes in how many different ways the n X n X n cube can be divided into subcubes.
%C A129668 One of the 1656 possible divisions of the 8 X 8 X 8 cube (42 of 1 X 1 X 1; 4 of 2 X 2 X 2; 2 of 3 X 3 X 3; and 6 of 4 X 4 X 4) solves the last unknown of the Hadwiger problem, n=54, found in 1973.
%C A129668 This sequence does not consider the way the cubes are arranged. - _Jon E. Schoenfield_, Nov 14 2014
%H A129668 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HadwigerProblem.html">Hadwiger Problem</a>
%H A129668 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CubeDissection.html">Cube Dissection</a>
%F A129668 a(n) <= A133042(n) = A000041(n)^3. - _David A. Corneth_, Nov 25 2017
%F A129668 a(n) <= A259792(n). - _R. J. Mathar_, Nov 27 2017
%e A129668 a(3) = 3 because the 3 X 3 X 3 cube can be divided into subcubes in 3 different ways: a single 3 X 3 X 3 cube, a 2 X 2 X 2 plus 19 1 X 1 X 1 cubes, or 27 1 X 1 X 1 cubes.
%e A129668 a(4) = 11 because the 4 X 4 X 4 cube can be divided into 11 different combinations of subcubes. The table below lists each of the 11 combinations and gives the number of ways those subcubes can be arranged:
%e A129668    (1) 64 1 X 1 X 1 cubes                       in   1 way
%e A129668    (2) 56 1 X 1 X 1 cubes and 1 2 X 2 X 2 cube  in  27 ways
%e A129668    (3) 48 1 X 1 X 1 cubes and 2 2 X 2 X 2 cubes in 193 ways
%e A129668    (4) 40 1 X 1 X 1 cubes and 3 2 X 2 X 2 cubes in 544 ways
%e A129668    (5) 32 1 X 1 X 1 cubes and 4 2 X 2 X 2 cubes in 707 ways
%e A129668    (6) 24 1 X 1 X 1 cubes and 5 2 X 2 X 2 cubes in 454 ways
%e A129668    (7) 16 1 X 1 X 1 cubes and 6 2 X 2 X 2 cubes in 142 ways
%e A129668    (8)  8 1 X 1 X 1 cubes and 7 2 X 2 X 2 cubes in  20 ways
%e A129668    (9)  8 2 X 2 X 2 cubes                       in   1 way
%e A129668   (10) 37 1 X 1 X 1 cubes and 1 3 X 3 X 3 cube  in   8 ways
%e A129668   (11)  1 4 X 4 X 4 cube                        in   1 way
%e A129668 The total number of arrangements is 2098 = A228267(4,4,4).
%Y A129668 Cf. A014544, A228267 (with multiplicity), A259792 (arithmetic instead of geometric partition).
%Y A129668 Cf. A034295 (same problem in 2 dimensions rather than 3).
%Y A129668 Cf. A000041, A133042.
%K A129668 hard,more,nonn,nice
%O A129668 1,2
%A A129668 _Sergio Pimentel_, May 02 2008, Jun 03 2008