Prove that one can pack precisely forty one 1x2x4 bricks into a 7x7x7 box.
Bonus Question:
Can precisely forty two 1x2x4 bricks be packed into this box?
If so, show how. If not, prove that it can’t be done.
The capacity of a cubic box of side 2n+1 for such bricks is at least n^3+(3n^2+1)/2; the proof is by partitioning the cube into parts that can be completely filled, leaving a remainder (2 for each pair of levels, plus 8m+5 for the odd 5 levels, where n=2m+1). So 41 bricks can fit into the box.
Equally, the capacity of a cubic box of side 2n+1 cannot exceed 1/2n(n+1)(2n+1). This would give a theoretical value of 42, but in fact no such packing exists. For odd cubes of side larger than 7, the precise maximum of bricks that can be packed remains an open question.
Quite hard for a D3 problem.
Edited on May 25, 2014, 8:33 am

Posted by broll
on 20140525 06:29:22 