Optimism is psychological room for a solution to exist.

When you are convinced nothing can be done, your mind stops searching, and the solution, if there is one, goes unfound. Optimism is partly just holding the space open where an answer might appear. It does not produce the answer; it prevents you from closing the door before looking.

If you have decided a problem is unsolvable, try approaching it once more from the assumption that something exists. Give it twenty minutes. The assumption alone changes what you consider.