Difference from liquid to solid items

Question

A shape's volume is the measure of how much three-dimensional space that shape occupies.

Challenge

Given six integers: [L,W,H] as length, width and height of one container and [l,w,h] as length, width and height of some random item.

You will need to calculate how many items would fit completely if the item was a liquid/fluid... and subtract how many items would also fit completely if the item was solid.

Note: Rotating the solid shape to fit empty space is required.

The input can be read from stdin, taken from args, can also be one or two lists/arrays, or any other valid way, but you can take in only six integers.

And the output should be one integer number: the difference.

Whole program or function are accepted.

Since this is code-golf shortest code in bytes wins!

Sample Input/Output

1) L=3  W=2  H=2  l=1  w=1  h=2
-> for item as liquid : 6 items fit completely
-> for item as solid  : 6 items fit completely
output: 0

2) L=1  W=8  H=3  l=3  w=4  h=2
-> for item as liquid : 1 item fit completely
-> for item as solid  : no items fit completely
output: 1

3) L=3  W=4  H=3  l=2  w=2  h=2
-> for item as liquid : 4 items fit completely
-> for item as solid  : 2 items fit completely
output: 2

4) L=6  W=6  H=6  l=1  w=5  h=6
-> for item as liquid : 7 items fit completely
-> for item as solid  : 7 items fit completely
output: 0

Draws

Just for illustration of how the solid items could fit:

Sample #3

Sample #4

Answer 1

Python 2.7 - 659 548 373 355 bytes

My first golf, but still very golfable.

The rotation thing didn't make it as easy as expected though.

import numpy as n
L,W,H,l,w,h=input();bx=n.zeros((L,W,H));s=0
q=(L*W*H)//(l*w*h);o=[[l,w,h],[l,h,w],[w,l,h],[w,h,l],[h,l,w],[h,w,l]];p=n.ndindex
for i in range(q):
 for a,b,c in p(L,W,H):
  for k in o:
   d=n.copy(bx)
   try:
    for x,y,z in p(k[0],k[1],k[2]):
     if d[a+x][b+y][c+z]:raise
     d[a+x][b+y][c+z]=1
    s+=1;bx=d
   except:pass
print q-s

But still a nice little brainteaser :D

**Edit: My colleague and I pondered a little more, and we found some more optimization potential

**Saved 18 byte thanks to CatsAreFluffy

Examples:

[6,6,6,1,5,6]
0

[3,4,3,2,2,2]
2