Building a box from smaller boxes John has 77 boxes each having dimensions 3x3x1. Is it possible for John to build one big box with dimensions 7x9x11?
I'm leaning towards no, but I would like others opinion.
 A: The answer is no; John can't even fill up the topmost $7\times 11\times 1$ slice of the $7\times 11\times 9$ box.  Consider just the top $7\times 11$ face of this box; look just at this face and ignore the rest of the box.  A solution to the problem would fill up this $7\times 11$ rectangle with large $3\times3$ rectangles and small $3\times 1$ rectangles.  But $7\times 11$ is not a multiple of $3$.
A: The volume of the big box is $V_B = 7\cdot 9 \cdot 11 = 693$, the total volume of the small boxes is $V_b = 77 \cdot 3 \cdot 3 \cdot 1 = 693$.
This means the volume of the small boxes is sufficient and we need to use all small boxes.
Let us try to model this problem Tetris style: 


*

*We have a base field, e.g. $7\times 9$, and need to drop all 77 small boxes over it.

*For each drop we have two decisions:


*

*where to put the center of the box $c = (c_x, c_y, c_z)$ over the base field $(c_x, c_y) \in I_x \times I_y$ with $I_x = \{ 1, \ldots, 7 \}$ and $I_y = \{ 1, \ldots, 9 \}$

*how to orientate the $3\times 3\times 1$ box. There seem only three feasible orientations: 


*

*a large $3\times 3$ side as base, like a pizza box ("O") 

*a small $3 \times 1$ side as base, orientated along the $x$-axis ("-")

*a small $3 \times 1$ side as base, orientated along the $y$-axis ("|")



*We loose if after the drop some part of the box sticks outside the big volume

*We win if we dropped all $77$ boxes without loosing.


This is a search space of $77\times 7 \times 9 \times 3 = 14553$ drop configurations. Not that much for a machine.
We could avoid the drop simulation and instead have $c_z$ as another choice. This would enlarge the search space to $77\times 7 \times 9 \times 11 \times 3 = 160083$ configurations. In both cases we need to check that boxes do not intersect.
This should be sufficient to code a solver which visits all configurations of the search space (brute force) and will answer the question by either listing feasible configurations or reporting that there is no solution.
Note: I submitted this before MJD published a counter argument.
