How should I frame a small void in my floor system to support subfloor?

Question

After tearing out the old subfloor in my half bathroom (due to water damage) I am replacing it with 3/4 in (2 cm) plywood. One side overhangs from the last joist about 9 in (22 cm) to the wall, and 11.5 in (29 cm) to the next joist. I plan to tile the floor with ceramics, so I need to brace that side somehow.

My plan was to attach three or four 11.5-inch 2x4s perpendicular to the joists and screw the plywood into those via 3-inch deck screws. My father also had a couple ideas:

• Cut a couple 4x4s and set them on the concrete below pillar-wise (unfastened), and screw the plywood into them
• Set a 2x10 on the concrete parallel to the joists, then screw the plywood into it -- this would require more of that black stuff underneath to raise it up to the other joists' level

If it's not obvious, neither of us are carpenters. I already have the 2x4s cut, but is my method unsatisfactory?

EDIT: Here are the three ideas in picture form (my idea, then my father's ideas):

Answer 1

Because of the short span I'd do the job with 2x4. 3" gold construction screws will do nicely.

1. Run a horizontal ledger along the far joist 3-1/2" down from the top. Fasten it with pairs of screws every 12" or so.
2. Run blocks across from the near joist on 16" centers (with one near the right end), end-screwing them twice at the near end and resting them on the ledger. Toenail them just once above the ledger to keep them in place. Your subfloor fasteners will do the lion's share of the work after that.

Consider using construction adhesive on everything to eliminate the chance of squeaks.

___________________________________________
                 far joist
__________o______________________________o_
      | o|                           | o|    <-- new ledger, down 3-1/2"
_____o|  |__________________________o|  |__
      |  |                           |  |
      |  |                           |  |
      |  |                           |  |
      |  |                           |  | <-- cross blocks
      |  |                           |  |
      |  |                           |  |
      |  |                           |  |
      |x |                           |x |
__________________________________________
       x         near joist           x
__________________________________________

  o = toenailed screws
  x = end-fastened screws

Answer 2

None of the ideas that you outlined are really ideal. Probably the best approach would be to cut some pieces of joist sized material (2 x 10s if I read your question properly) to be used as blocking between the open joist and the one down under that wall. These would be cut to match the spacing between those two joists. You can use metal joist hangers to secure them in place. I would space these equivalent to your normal joist spacing.

This picture shows the idea where you would place these.

Another alternative is to use some joist sized material cut to length and just lay it up against the outer joist that is under that wall and simply screw it through the face into that covered joist. It may take several iterations of this to being out the top of these sistered joists so that the top extends beyond the wall surface. This alternative may be easier so as to avoid difficulty with the electrical wire running through that area unless the wire is already below the bottom of the existing joists.