Can you solve it? Prisoners and boxes

Using brain cells in prison cells

UPDATE: Read the solution here

Today's puzzle involves two prisoners. Let's call them Piper and Alex. You will be asked to find a strategy that wins them their freedom.

Puzzles about prison escape strategies are a relatively new genre of mathematical challenge, only a few decades old. They emerged from computer science. A prison - a place where access to information is limited - is a perfect venue for puzzles about the efficient communication of information.