Cover a single cube with FIVE identical cube nets

Question

Start with five identical cubes:

Your challenge:

1. Cut and unwrap all five cubes into five identical cube nets.
2. Show how to re-fold these five cube nets to form the surface of a single larger cube, with five times the surface area of one of the original cubes.

Rules and clarifications:

• The original five cubes must be the same size.
• The cube nets must be made by cutting the original cubes along their edges only.
• The five cube net shapes must be identical to each other. (Mirror images do not count as identical.)
• The five cube nets must cover the entire surface of the final cube, with no gaps and no overlaps.
• The edges of the cube net shapes do not need to line up with the edges of the final cube.
• I am aware of one solution, but I can't guarantee it's the only one. Any solution that meets the above requirements is valid.

Note:

This is a variation on my previous puzzle: Turn two cubes into one! This new puzzle is very similar to that one in terms of how the puzzle is worded, but the solution is quite different, and was (for me, at least) much harder to find. (Honestly, when I started exploring this one I was kind of expecting to make a puzzle where the challenge is: "Find a clever way to prove this thing is impossible".)

Answer 1

Only one of the 11 normal cube nets allows the large cube to be formed – the cross-shaped one – and the solution is unique: The centre cell of each cross is marked with thin lines above.

The solution was found and proved to be unique using an exact cover program on the tiling of the larger cube, which has to have side length $\sqrt5$. I started with six "rings", where each ring contains the squares visited if you start from some square and keep moving forwards: From this I found for each one-sided net (as required by the problem rules; the large cube tiling is chiral) all admissible ways of placing it on the large cube tiling, using a turtle graphics-like approach to specify the cube nets (aided by the "ring decomposition" of the squares from above). These ways in turn form the input for the corresponding exact cover problem.

All nets but the cross-shaped one led to zero solutions; the cross-shaped one led to 24. Since the tiled large cube has 24 symmetries and the solution above is totally asymmetric, however, the above solution is unique.