Stuffing Puzzles – Xmas Stocking

I’ve seen references to the Xmas Stocking puzzle recently that caused my mind to wander a bit.  Since all the pieces were identical and packed in a fairly small box, I was wondering if I could solve it without having the puzzle.  After several minutes of determining how the pieces would need to be oriented due to the opening and rotation restrictions, I was able to visualize some assemblies but not how the pieces could be placed in the box.  I eventually resorted to using a piece of paper to map out the possible assemblies with no further luck.  It’s possible that this is one of those puzzles that requires some assistance like larger tolerances or beveling.  I was about to move on when I remembered that I had received some LEGOized hand-me-down puzzles from a generous soul on the Mechanical Puzzle Discord (MPD) server and sure enough, there it was.

Xmas Stocking is a 3D restricted-opening packing puzzle designed by Osanori Yamamoto.  It consists of 5 identical Z tetracubes that have to be placed in a 2x3x4 box with 2.5 voxels on the top closed off.  The triangular half-square is there to restrict pieces from moving through that square while forcing the voxel underneath to be occupied – assuming that the opening needs to appear completely filled.  I was unable to find verification of that requirement, but these types of puzzles seem to require that and it turned out to be the case for this puzzle.

Most of the 3D LEGO packing puzzles that I recently received came solved.  However, since they are made from LEGO bricks, you can simply rip off the bottom and dump the pieces without seeing how they are arranged or how they come out.   The LEGO bricks used to make the box also included portholes to view the interior of the box, but they are not needed to solve the puzzle.  

Being made from LEGO bricks, you know that there are no rotations that can occur with loose tolerances or bevels unless very specific pieces are used or they are doctored in some way.  Armed with that assumption, I tested the pieces and could get no further than my pencil and paper analysis.  Yes, of course I realized that I was operating with a bad assumption somewhere but was having a difficult time identifying what it was.  And then I dropped a single piece in the box and it bounced it’s way into an orientation that I had thought was impossible.  Apparently, the Z tetracube has a nice property that gives it a possible rotation that I had not thought of.  Now that the secret was out, it only took a couple of minutes to get all 5 Z tetracubes nestled within the box.

I don’t know if it was intentional, but the LEGO version did have the interesting property of still being apparently filled even with all those holes.

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