title: Estimate Feasibility of filling Main-Cube interior surfaces with some sub-cubes without Volume Interference
body: <p>I have a stupid question but it's really hard I think to compute and it's part of my project to estimate the feasibility of layout these small boxes on the interior surfaces of the big empty box("Main-Cube") at the beginning of process ... So in my problem I have a big extruded cut cube (like empty box - that we called it "Main-Cube" as shown in the below fig ) and variable quantity  small solid boxes (quantity of these small boxes is predefined by end user as input data and we called them "sub-cube" ) that stick to interior surfaces ( mate/projection/connected to surface) ... So I want to know when this problem Certainly has no answer (because of crowding small boxes on each 6 side of main cube interior surfaces ) and so won't let the end user to start computing at first time and tell the user that this problem Certainly can't have any feasible answer ( feasible answer means has no interference/violation in volume) so the user won't waste time for further computation .</p>
  
  <p>for example the Main-Cube dimensions is 1000mm x 1000mm x 1000mm
  I think the total Main-cube volume is (without considering 10mm for Mantle/thickness of cube) is Approx : 1 Cubic meters or 1,000,000,000 Cubic millimeters 
  I also find the total small box volume so the fraction of : 
  Total Sum of small boxes Volume / Main-cube total volume</p>
  
  <p>is not really Useful ! 
  I also try the sum of minimum area for each small boxes and then calculation this fraction :
  Sum of Min All small boxes / 6 x (1000mm x 1000mm)</p>
  
  <p>and because of common edges of Main cube it can't be useful too .
  So any Idea or concept accepted :) that can Ensure that with this used area definitely no solution can find for this problem unless reducing some small box and repeat the process check ...</p>
  
  <p><a href="https://i.stack.imgur.com/WIifz.png" rel="nofollow noreferrer">Crowding scenario</a></p>
  
  <p><a href="https://i.stack.imgur.com/voCs8.jpg" rel="nofollow noreferrer">Low dispersion for simple scenario</a></p>
  
  <p>P.S : As you seen in the below table (contain small and big box dimensions) the results said that roughly 44% interior surfaces area and 26% volume is filled (or used by 90 small boxes) how ever now the problem rarely has feasible solution and it's really hard to say if we increase amount from 90 boxes to 100 We can find one feasible solution but 26% or 44% don't represent this!</p>
  
  <p><a href="https://i.stack.imgur.com/UPUgg.png" rel="nofollow noreferrer">Specifications Table</a></p>