Help Star Trek’s Lieutenant Uhura Overcome Astronomical Odds

I knew this problem as the '100 prisoners' problem.

The warden offers 100 prisoners a chance at freedom - if every one of them can find their key, randomly placed in 100 numbered boxes. The keys are numbered (honestly) as well. Each prisoner separately can make 50 attempts (open up to 50 boxes sequentially).

Every prisoner must find their own key.

What are the odds, all of them can succeed? Surprisingly high - around 50%. If they follow the right algorithm.

My insight was a game we used to play in Scouts - everybody face the center of the room in a crowd, reach across and grab two hands at random. Then try to untangle all the loops formed without letting go.

Clearly only loops can be formed, and clearly more than one can be formed.

The odds of multiple loops is high - in fact, the odds that no loop is larger than half the participants is high.

That's the critical insight that allows every 'prisoner' to succeed. Sometimes. Maybe half the time.