I find keeping an open mind can result in solutions that are easy to understand and make sense, but sadly to me only. The problem is, this approach can only produce solutions that nobody is willing to consider.
Awareness and purpose make no mathematical sense, as our laws of physics seem to prevent them, and the Chinese Room argument is a strong argument that cannot be overcome.
Solution making it appear impossible
Expected assumption, possibly something similar to this, "consciousness is explained mathematically, but nobody has any mathematical explanation because it is so extremely complicated, that nobody understand.".
Simple solution nobody will consider
Awareness and purpose make no mathematical sense because they are not mathematical, they are the opposite of mathematical. The opposite in every way. Now we have physical laws that literally define consciousness and purpose.
I could write pages of logical arguments backing up my claim. No matter what though, I've never received a single comment logically supporting or contradicting my claims. I have had an argument that suggests I am correct being used against me before though, which doesn't even make sense. Refining a non-conventional theory, means you can be the only one contradicting it with relevant logical arguments.