12 piece cube packing puzzle

Question

Consider the following hexacube (made from 6 unit cubes):

GOAL:

Pack a 3 x 3 x 3 cube using three of these hexacubes plus nine unit cubes.

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This puzzle comes from: https://puzzlewillbeplayed.com/333/Best/12/

Answer 1

What a great puzzle!

For me the key was to notice that

if you try to put the apparent hexacube corners (the point from which the three "prongs" radiate) at the corners of the 3x3x3 cube,

you will quickly run out of corners. Since there is only one other way (plus a zillion symmetries) to place the hexacube, this means that we know the location of at least one hexacube. Using that as a starting point, the next step was to fiddle around a bit, and realise

you actually need to place every hexacube in that way.

You can point the short prong of each piece in either direction, and assemble the pieces in either a clockwise or anticlockwise manner, but other than that, it seems the solution is unique.

EDIT: @JaapScherphuis points out in the comments, that if you account for all the symmetries, there are only two distinct configurations: One with 3-fold symmetry where all the short prongs point the same way, and another, where one short prong points in the opposite direction from the other two. The picture below shows the latter variety.

To finish off, I then spent a couple of hours learning how to do 3D modeling with a camera flying in a loop, and after that, I only needed to figure out how to squeeze an animated gif to fit within the site's 2 megabyte image size limit. But maybe it was all worth it:

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Post scriptum

As to some particular reasons why I really like this puzzle design:

• The pattern is pretty, and surprisingly tricky to find
• There is a path of logical deductions that will lead to the solution (see the answer above)
• The extra unit cubes give nothing away, but fill all the holes, immediately ruling out any non-orthogonal ideas
• There's room in the cube so that every piece could have had another short prong opposite the existing one. Adding that prong would make everything even prettier and more symmetrical, and furtherhencemorth, the alternative solution would disappear. That extra prong would immediately give away the solution though, so the designer did an excellent job of balancing aesthetics with the challenge level.

Answer 2

Leaving out small cubes:

    top

     A A .
     . A B
     . A .

     middle

     C A .
     C . B
     C C C

     bottom

     . A .
     B B B
     C . B