For some reason,
Alatar, your amusing suggestions remind me of another disappearing and reappearing character created by another Oxford professor - Lewis Carroll's Cheshire Cat! Perhaps Sauron did it similarly? However, it wouldn't be his grin that remained longest, but his Eye - that explains all!
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'Mercy!' cried Gandalf. 'If the giving of information is to be the cure of your inquisitiveness, I shall spend all the rest of my days in answering you. What more do you want to know?' 'The whole history of Middle-earth..
.'