# 12 piece cube packing puzzle

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.

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

• Interesting: the puzzle gets easier if you ROT13: nqq n 7gu phor gb gur funcr. .. va gur evtug cynpr bs pbhefr. Commented Jul 5, 2023 at 6:37
• so essentially the puzzle is, how to pack three of these into a 3x3x3 cube. Commented Jul 5, 2023 at 9:03
• @justhalf You are right. Commented Jul 5, 2023 at 9:15
• @justhalf the unit cubes rule out any non-orthogonal placement ideas without giving anything else away, one of the many things I like about this puzzle.
– Bass
Commented Jul 5, 2023 at 12:10
• ah, makes sense @Bass, even though I'm not sure it's possible in this case to put the piece non-aligned with the unit cubes. Commented Jul 5, 2023 at 16:10

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:

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.
• Nice animation! Up to symmetry, this puzzle has only 2 solutions. If you flip the blue piece around, you get the other solution which has 3-fold symmetry around a main diagonal of the cube. Commented Jul 5, 2023 at 13:16
• Thanks, and I agree about the solution count. The "up to symmetry" itself is a bit tricky though, because while the hexacube piece is mirror symmetrical, and so of course is a cube, the solution pattern is indeed chiral: if you look at a cube corner where all the three pieces meet, in the animation the pieces always form a counterclockwise pattern, which means that in the mirror image of the solution that pattern will go the other way.
– Bass
Commented Jul 5, 2023 at 13:37
• 3D modeling might not be necessary if you have Minecraft :)
– iBug
Commented Jul 7, 2023 at 6:22
• @iBug Are you trying to claim Minecraft doesn't count as 3D modeling? :-O
– Bass
Commented Jul 7, 2023 at 12:10

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