the total count of cubes increases as more and more cubes are added around an existing cube. each "level" is defined as completing the addition of an entire layer of concentric cubes that surround the previous level.
level 0 is no cubes at all
level 1 is ONE cube
level 2 surrounds level 1 with 8 more, 9 total cubes in each of 3 planes, total = 27
level 3 surr the 9 in l-2 /w 16 more, total 25 cubes in each of 5 planes, total = 125
level 4 surr the 25 in l-3 /w 24 more, total 49 cubes in each of 7 planes, total = 343
level 5 surr the 49 in l-4 /w 32 more, total 81 cubes in each of 9 planes, total = 729
these numbers, 0, 1, 27, 125, 343, 729 each represent the total count of cubes at any given concentric level. let us call this CC_T_n which stands for the Count of Cubes _ Total _ at level n.
i figured out that:
CC_T_n = ((n * 2) - 1) ^ 3
we are interested in the count of cubes required to surround a smaller block of cubes; EXACTLY what i'm doin' right here! let us call this CC_OL_n which stands for the Count of Cubes _ Outer Layer _ at level n.
if you think about it, this will equal the total count of cubes in any block at level n minus the total count of cubes in the block at level n - 1; why? because, the total count at level n - 1 is the count of blocks that the count at level n must cover!
and so, CC_OL_n = CC_T_n - CC_T_n-1, which equals:
(((n * 2) - 1) ^ 3) - ((((n - 1) * 2) - 1) ^ 3)
working this though the consecutive levels, i got the following results:
l-0 CC_T_0 = 0
l-1 CC_T_1 = 1, CC_OL_1 = 1
l-2 CC_T_2 = 27, CC_OL_2 = 26
l-3 CC_T_3 = 125, CC_OL_3 = 98
l-4 CC_T_4 = 343, CC_OL_4 = 218
l-5 CC_T_5 = 729, CC_OL_5 = 386
l-6 CC_T_6 = 1331, CC_OL_6 = 602
l-7 CC_T_7 = 2197, CC_OL_7 = 866
i typed the following string into google:
and i found a web site that categorizes various numeric sequences. it had a reference to a guy named Xavier Acloque with a comment "Numbers of cubes needed to completely "cover" another cube."