Some unit cubes can together form a structure, such as in the cover image above, or such as this
problem. The cubes are ordered just as if they are voxels
, and gravity takes effect downward (so no cubes can be floating), but otherwise there's no restriction.
This time, you don't know the structure! You left the structure for lunch break, and when you returned, it's gone. You managed to take a picture of the structure, though. In fact, you don't only have one picture; you have three. You took:
- one from the front of the structure (top side is the direction up, right side is the direction right),
- one from the right side of the structure (top side is the direction up, right side is the direction back), and
- one from above of the structure, looking down (top side is the direction back, right side is the direction right).
However, much to your surprise, when you look at the pictures they are all exactly the same!
If \(m\) and \(M\) are respectively the minimum and maximum possible number of cubes in the structure, determine the value of \(M-m\).
Image shamelessly stolen from here.