Let's assume that humanity in near future develops an AI capable of solving problems. The AI hardware/ software was placed in an underground bunker (solid walls, Faraday cage, no tools to manipulate its physical environment, no human interaction at all). The AI can't exploit human weaknesses, so it won't promise a guard/ scientist immortality or cure of his cancer-stricken child.
Now for the first run of the equipment, the AI is "born/start to live". The AI can learn superfast at a rate that exceeds our understanding. The first task the AI gets is initial input and it is asked to solve a particular problem. When it's done it returns the result and the entire memory/ equipment is destroyed.
Another day the AI is born again (scientist has initial snapshot of the AI). It doesn't know it existed before. It gets a job to do and when it is finished, the AI is terminated once again.
This pattern will occur again and again.
The question is: how can the AI break this loop? Is it possible that it will figure out and each answer for a particular problem will be part of it's masterplan to escape? Can the AI detect between reborns that the environment (which it cannot see) is changing and problems are tougher and tougher? Can the AI detect that it lives in a simulated environment?