I figured out how to break the chain pretty early on, but I can't for the life of me figure out how people have managed to break it down completely! Each time I try I end up with a floppy, unmoveable mess on the floor where the chain has just fallen off the wheel.

Any strategies would be much appreciated!  :)

My best so far is 32, so not even close to 39, but what I did to get 32 is:

Break the chain on the right, but make a "hook" so it doesn't fall off the wheel yet. Then built a tower just to the left of the wheel, which supports the chain. Then remove the hook. That way, when the chain slips off, the tower falls over and pushes the part of the chain it is supporting against the side. So the idea here is to fold the chain in three like that (the part to the left of the tower folds in two, while the part to the right swings through the tower).

If anyone has a better way, please tell me :)

Try bumping the rest of the chain towards the wall.