Box stacking problem Real world problem alert: I am moving from my house to another one, and the problem below arised when I tried to fit some little boxes of various shapes into a large box:
We are given a positive integer $n$ and $3$-tuples consisting of positive rationals $(w_i, l_i, h_i)$ for $i \in \{1,\ldots,n\}$. For each $i$, the tuple $(w_i, l_i, h_i)$ represents a box of width $w_i$, length $l_i$, and height $h_i$.
We want to put these $n$ boxes into one big box of rational dimensions $(W, L, H)$ such that the volume $W\cdot L\cdot H$ of the big box is minimal.
Is the problem of finding (one possible choice of) $(W,L,H)$, such that $WLH$ is minimized, computable?
EDIT. Thanks to Reid Barton and Tony Huynh for their comments below. The boxes can be rotated arbitrarily.
 A: Yes, this is computable. And the infimum is attained. To see this, observe you can express a placement of your boxes by giving the coordinates of each box corner together with the angles. Then this placement is a valid packing if a (big) collection of simple inequalities are satisfied. It's clear that there are only finitely many combinatorially different placements (as defined, say, by extending all the bounding planes and looking at the combinatorial structure of the corresponding arrangement of planes) and hence one can simply minimise (continuously, hence attaining the minimum) over each structure. This is (a) possible, actually fairly easily, but Tarski's theorem is easier to quote, and (b) not in any way an efficient algorithm..!
A: This is not an answer, but instead addresses a related problem:
The problem of finding a minimum area rectangle that encloses a given set
of rectangles, without rotation (i.e., all sides parallel to Cartesian $x$ & $y$ axes)
is NP-hard. 

Huang, Eric, and Richard E. Korf. "Optimal packing of high-precision rectangles." In Twenty-Fifth AAAI Conference on Artificial Intelligence. 2011.
  PDF download.
Korf, Richard E. "Optimal Rectangle Packing: Initial Results." In ICAPS, pp. 287-295. 2003. PDF download.


          


          

Korf (2003): Optimal packing of the first 22 squares.


