For small n, the problem of orthogonally packing consecutively sized integer cubes 1x1x1 - nxnxn inside the smallest integer cube CxCxC is trivial. By inspection, the sizes of two largest cubes n and n-1 must bound C, so C = 2n - 1 for small n. Also by inspection, you can see that by putting the largest cubes in the 8 corners of C, the 9th largest cube must be placed next to 2 other cubes, best case the 7th and 8th largest cubes, so C is bounded by (n-6) + (n-7) + (n-8) = 3n - 21. This becomes the limitation, in other words, 2n-1 = 3n-21 for n=20, and 2n - 1 < 3n - 21, for n > 20. n=20 can be said to be a transition point between two bounding equations for C.
Again, this becomes a fairly trivial exercise until n=69 (see http://demonstrations.wolfram.com/CubePacking/ ) with a packing density 90%+.
Q1: Is packing cubes 1-70 still bounded by 3n - 21 ? In other words, can cubes 1-70 orthogonally pack into 189x189x189? Orthogonally means the x,y and z axes of all the cubes point in the same direction, no rotation or tilting allowed.
Q2: Presumably, for n=70 or a value not much larger, C = 4n - k. Is this true, or is there some other bounding equation? If so, what is k, and can you make an argument similar for 3n-21 why k has the value it does? In other words, can you show that at minimum 4 squares n-a1, n-a2, n-a3, n-a4 must be next to each other, defining a new lower limit for C ?
Q3: What is the smallest n and C to give a packing density >99%? I can create this for somewhat large n and an inefficient packing strategy. I am looking for a better packing strategy and smaller n.
Q4: Presumably, as n approaches infinity, packing density approaches 100%. Can you make an argument for this? My most naïve shelf packing approach did not converge to 100%. Can your approach?
Q5: Packing density after transitioning from C=2n-1 to C=3n-21, at n=21, is approx. 72%. Counterintuitively, packing density goes down as n goes from 21 - 29, and does not return to > 75% until n>=45. Will similar behavior be seen after the next transition to C=4n-k?
Q6: Can further transition points be demonstrated for C=5n-k2, C=6n-k3, etc.?