Your comment and neq1's intuition pump prompted me to create the following reformulation of the problem without amnesia:

I flip a coin hidden from you, then ask you to name a number. If the coin came up heads, I write your answer into my little notebook (which, incidentally, is all you care about). If it came up tails, I write it in the notebook twice.

When the problem is put this way, it's clear that the answer hinges on how exactly you care about my notebook. Should it matter to us how many times we express our credence in something?