| hard | discrete |
Can you pack 53 bricks of dimensions 1x1x4 into a 6x6x6 box? The faces of the bricks are parallel to the faces of the box
See its cousin called Domino Covering in which two opposite corners of a checkerboard are removed.
No!
Divide the volume of 6x6x6 box into 2x2x2 mini cubes. Imagine each of these mini cubes is either fully red or fully blue such that it forms a 3D checkerboard pattern on the box. This will restrict 14-13 restriction on cube colors, say 14 blue and 13 red. Now, putting bricks into box, parallel to faces, each brick will be half blue and half red, so 52 bricks fill all the red cubes and there is no way to place a 53rd brick.