About This Puzzle
The Hexagonal Bipyramid Imaginary Cube (abbreviated H) is the solid obtained by gluing two regular hexagonal pyramids together at their bases, as shown below.
The Triangular Antiprismoid Imaginary Cube (abbreviated T) is the shape produced by cutting away the vertices around one end‑face of a regular triangular prism, as in the figure below.
Both H and T fit perfectly inside a unit cube, and from every face of that cube they appear as perfect squares. Solids with this property are called Imaginary Cubes.
The puzzle Imaginary Cube Puzzle 3H = 6T asks you to pack three H pieces and six T pieces into a box whose edge length is twice that of the unit cube. Since the box is twice as large, fitting eight pieces is trivial—but here we must fit nine. How can all nine pieces be packed?
Solving the packing itself is entertaining, but the finished configuration also lets you feel the beauty of the underlying mathematics. Even if you do not have the physical puzzle, you can still enjoy it by reading the Hints & Solutions and the detailed Explanation. Once you understand the mechanism, handling the real pieces becomes even more satisfying.
Imaginary Cube Puzzle 3H = 6T
This mathematical puzzle challenges you to pack nine pieces of two different shapes into a cubic box. Placing eight pieces is easy—what about the ninth?
Although the task seems impossible at first, a remarkably elegant solution exists. When the final piece snaps into place, the beauty of the completed shape is striking.
But the story does not end there. As you admire the result you may wonder, “Why does everything fit so perfectly?” or “Is it really flush with the interior of the box?” This is the second puzzle.
Elementary mathematics provides the answer: you can rigorously prove that the assembly fits exactly, and the proof is accessible at junior‑ high‑school level. Playing with the two solids also reveals that they possess beautiful tiling properties—H and T are indeed “special” polyhedra.
Enjoy exploring both puzzles to the very end!
- Set contents: 9 puzzle pieces, large case ×1, small case ×1, booklet
- Recommended age: 8 and up
- Price: ¥2,750 (incl. tax)
- Product code: IMCB001
Development History
- 2021 – “Imaginary Cube Puzzle 3H = 6T” (standard edition), Image Mission Mokkyosha (plastic pieces, fit inside a 3 cm cube)
- 2012 – “Imaginary Cube Puzzle 3H = 6T” (booklet + puzzle), Kyoto University Museum MUSEP (Active KEI) – out of print (wooden pieces, fit inside a 4 cm cube)