216

3³ + 4³ + 5³ = 6³ = 216

8 pieces are required to construct a puzzle that can be assembled as a 6×6×6 cube and reassembled into the three smaller cubes. The 6 "slices" of the puzzle would appear as follows, with letters a through h representing the eight puzzle pieces:

  f f f f f h    f f f f f h    f f b b b b
  f f f f f e    f f f f f e    f f b b b b
  f f f f f e    f f f f f e    f f f b b b
  g g g e e e    g g g e e e    g g g b b b
  g g g e e e    g g g e e e    g g g e e e
  g g g e e e    g g g e e e    g g g e e e
    slice-1        slice-2        slice-3

  f f b b b b    a a b b b b    a a b b b b
  f f b b b b    a a b b b b    a a b b b b
  c c b b b b    c c b b b b    c c b b b b
  c c b b b b    c c b b b b    c c b b b b
  c c c e e e    c c c e e e    c c c d d d
  c c c e e e    c c c e e e    c c c d d d
    slice-4        slice-5        slice-6

216 was given spiritual significance in ancient times because it is the product of the first two cubes (they did not count 1³ as a "cube"): 2³ × 3³ = 216

Numerologists relate 216 to 666 because 6×6×6 = 216