To hang one’s head, is a lowering of your chin and your gaze, generally in shame. Right?
Does this strike you as an odd phrase? Would it not be better to droop one’s head? Hanging sounds an awful lot like a hanging. And outside of hangman, there’s been no good hanging. Ever.
Hang your head in shame! And pretend you have a hemp necktie!
And seriously, can hangman move onto a cool geometric shape or something? Not only would it be less morbid, it would also be more consistent.
So remember: it’s better to find a place to hang your hat than hang your head.