# Seven octahedral nets to cover an octahedron

After solving Cover a single cube with FIVE identical cube nets I had the idea for this puzzle, which may be regarded as a natural generalisation to triangular grids.

Find two different nets, A and B, of a regular octahedron such that these two collections of nets can each be folded into the surface of a single regular octahedron with seven times the surface area of the original octahedron:

• 3 copies of A and 4 of B
• 6 copies of A and 1 of B

Restrictions from the five-cubes-to-cube puzzle apply analogously:

• A and B must be formed by cutting along the original octahedron's edges (so they are octiamonds). All small nets are of the same size.
• The nets are one-sided. Suppose one side of each net is painted, then all copies of A must be identical without flipping when the painted side is up, and similarly for B. The large octahedra formed by both net collections must show only painted faces.
• The large octahedra must have no gaps or overlaps in them.

As a hint, A and B differ by only one triangle.

A and B are also totally asymmetric (not rotation/reflection-symmetric).

If one triangle is removed from net A the remaining heptiamond has the symmetries of a triskelion. Net B can be decomposed into two Pac-Man-shaped tetriamonds.

• My goodness this is a tough one! Nov 17, 2022 at 9:12
• @plasticinsect I found no solutions using only one kind of net or a net and its mirror image. This was a choice pick over all 2-net combinations. Nov 17, 2022 at 9:34

Here is already a 6A + 1B covering.

With the same shapes there is a 2A + 5B covering.

[edit: Except that, as Parcly mentions, it doesn't satisfy the one-sided rule...]

But I don't know about 3A + 4B...

Update: I found a second covering with the same shapes, that respects chirality But it is 5A + 2B. And since it is the result of a computer search it seems these 2 shapes won't do.

Finally

The new hints restrict a lot what shape to look for. There were just two possible pairs, (i.e. 2 orientations) and only one of them yields a solution. The solutions happen to be composed of 6A + 1B

and 3A + 4B.

Below you can find the OP's solutions. It is bound to be the same solutions but in a very different presentation. (From OP) The code that found this tiling pair can be found here.

• Funny, I recall answering that, but I don't know where. It is a basic paint program called acorn. It is probably less advanced than you think. Dec 6, 2022 at 22:13
• Ah, thank you, I'll check it out. Dec 6, 2022 at 23:49