How many legal positions are there on a Rubik's cube with two pieces glued together? Legal here means reachable from the solved state without separating the two glued pieces. Answer depends on the type of pieces glued together (egde+corner or edge+center). Is it just all normally legal positions where the two pieces are adjacent?
 A: It indeed turns out that bandaging just two pieces together does not inhibit the amount of mixing the other pieces can get, so the number of positions of this bandaged cube is indeed the simply the number of cube positions with those two pieces adjacent.
In fact, if you take three faces adjacent to a corner, and mix the cube using only moves of those three faces, then the moving pieces can be fully mixed. That is to say, you can bandage a whole 2x2x2 block, and the resulting puzzle has the same number of positions as a normal cube with that block solved. To put it another way, if you solve the whole cube by first solving a 2x2x2 block, then you won't need to temporarily break that block when solving the remainder. This does not happen if you choose three faces, two of which are opposite. With turns of three such faces you cannot flip any edge pieces.
To prove this kind of thing you can either explicitly work out a solving method (usually be placing the bandaged piece correctly first), or you can use a program such as GAP to model the group and calculate its size. To model the group you will have to consider the bandaged piece as being fixed, and all the other pieces including face centres as moving pieces. That group is generated by moves of any face layer and middle layer that does not affect the bandaged piece.
Things get very complicated when you have several groups of bandaged pieces. When they move around in different arrangements, some moves get blocked. The puzzle can no longer be modelled as a group generated by single moves of the layers. Even working out the number of positions an unstickered bandaged cube has becomes tricky, and generally boils down to trying all possibilities (e.g. by computer search). 
