5x5x5 Rubik's Cube
I solved it. My wife scrambled it at my request on Thursday night. I got a lot done Thursday but didn't have much time for it on Friday and Saturday, but finished it up Saturday evening.The procedure is to pair up the 2nd and 4th edgies first, which may involve parity problems of the type I saw on the 4x4x4 cube. Then I used 3x3x3 routines to solve the single corner problem combined with the 2nd and 4th edgie pairs matched with the absolute (unchanging) center of each side. That left the 8 centers on each side and the 3rd or middle edgie. There are two types of movable centers: sides and corners of the little 3x3 patches of centers. But I used the same method, borrowed from the 4x4x4 cube solution, to do both. Last to fix was the middle edgies. Several were already in the right place after step one, when I filled them in when I could. I used a basic routine from the 3x3x3 cube to move them around. And I was able to used the "Rubik" routine from the 3x3x3 cube to flip the last pair, but it messed up some of the 3x3 patches of center pieces. So the last step was to clean up those center pieces.
It probably took me 4 hours, but I must have wasted an hour due to improperly applying my old routines, routines I know by habit on the 3x3x3 cube which had to be applied slowly and meticulously on the 5x5x5, both due to the size and because sometimes I grabbed two layers and sometimes I grabbed one layer (on purpose).
Fun challenge!