Solution

Gap‑Free Packing

Step 1
Begin by placing an H‑piece. Leave a space in the very center of the box.

Solution Step 1

Step 2
Add a T‑piece next to the H so that the two meet along their narrow isosceles‑triangle faces.

Solution Step 2

Step 3
Place another T on the opposite side of the H.

Solution Step 3

Step 4
Fill the remaining positions in the bottom layer with T‑pieces. This time they touch the neighboring pieces along an edge—two edges of a large equilateral triangle. A cavity now appears in the center, exactly matching the pointed tip of a T‑piece.

Solution Step 4

Step 5
Insert a T into the central cavity.

Solution Step 5

Step 6
Continue adding pieces so that adjoining faces always match.

Solution Step 6

Step 7
Keep matching identical faces.

Solution Step 7

Step 8
Match the remaining faces in the same way.

Solution Step 8

Step 9
All nine pieces are now in place—puzzle complete!

Solution Step 9

You can start with other placements in Step 1 and still reach a solution. However, every gap‑free arrangement is equivalent to this one by simply rotating the box.

Solutions with Gaps

Many packings leave small gaps between pieces, yet every known solution of this type still places a T‑piece at the center, as in Step 5.

Counting only the solutions that put a T in the middle, there are 512 distinct packings (ignoring horizontal rotations of the box, but treating mirror images as different).
If mirror‑image solutions are identified, the number drops to 276.
Identifying solutions that coincide under any 3‑D rotation of the box leaves just 52.

You can inspect the calculation in this Python program, or run it in Google Colab. It is curious—though probably coincidental—that 512 is a power of 2.

Yoshiaki Araki of the Tessellation Design Association has created a diagram of 136 configurations—the 276 solutions with the central T fixed, modulo mirror symmetry.