Bandaged Cube (Meffert's)
Bandaging a cube means sticking together some of the cubies. This creates some interesting sticker patterns and, because the number of faces that you can turn is restricted, the cube is quite tricky to solve.
The image above shows a Meffert's bandaged cube which uses a particular bandaging pattern. The tiles are stuck onto a standard 3x3x3 cube and, if removed, leave you with a normal cube.
Production runs of this puzzle by Meffert come few and far between. Whilst waiting for these to come back in stock, I made my own. I glued the cubies together in the correct pattern - taking as much care as I could. I then made some stickers (well, got a more arty friend to do that) out of vinyl sheets that can be ordered online. This made a perfectly workable bandaged cube. Be careful if you do this - not just with the glue, obviously. If you stick the wrong cubies together, you have a completely different puzzle.
One interesting way to do this is to stick lego bricks on all of the cubies and then use more bricks to bandage the cubes together. This avoids the need to glue any cubies and allows you to change the bandaging pattern at any time.
The bandaged cube is one of my favourite puzzles. Your friends struggle to scramble the thing for you - there is only only one piece that isn't connected to at least one other (it is referred to as the free corner) - a face will only turn if it contains this corner - even then, there are some restrictions.
Get your bandaged cube scrambles here.