joe_meadmaker wrote on Dec 7
th, 2019 at 10:09am:
That makes me think that the puzzle should have a significantly higher number than the Rubik's Cube.
Exactly.
So here it is:
Case 1 (as in my first post): Corners, edges and middle parts can not be distinguished.
So for the first piece you have 2 sides, 4 rotations and 64 positions to choose from.
As you have to multiply the number of choices.
2*4*64 = 512 possible choices for the first piece.
For the second its: 2*4*63 choices.
Therefore for all pieces: (2*4*64)*(2*4*63)*(2*4*62)* .... (2*4*1) choices.
This can be simplified to: (2*4)^64*64! which you can type into e.g. wolframalpha.com to retrieve: ~ 8*10^146 which is an 8 with 146 zeros after it.
Case 2: you can distinguish the corners, edges and middle pieces.
For the 4 corners you have: (2*4)*(2*3)*... = 2^4 * 4! = 384 choices.
For the 24 edges you have: (2*24)*(2*23)*... = 2^24 * 24! = ~1*10^31 choices
For the 36 middle pieces you have: (2*4*36)*(2*4*35)*... = 8^36 * 36! = ~1*10^74 choices.
For the total number of choices you have to multiply everything again. So:
2^4 * 4! * 2^24 * 24! * 8^36 * 36! = ~5*10^107
The extra mile:
For the above it was assumed, that overall rotation and flips are different, i.e. rotating the final puzzle gives a different solution, although all pieces are connected in the same way.
If we want to account for this, we have to divide each "total number of choices" (~ 8*10^146 and ~5*10^107) by (2*4) = 8.
For the maximal number of attempts needed to solve it, you need to define an algorithm, i.e. instructions on how to solve it. Then the question can be answered.
The stupid method: put all pieces on 8 by 8 square and see if they fit together (Hey, it's a legit way to solve it

).
You need maximally ~ 8*10^146 attempts, so the number from case 1 above. Because then you have literally tried all possible ways to lay them out.