This puzzle is a variant of the famous “prisoners and the light bulb” problem, where 50 wise men are tested not by riddles, but by coordination under uncertainty.

The Setup

  • 50 wise men are imprisoned separately.
  • Outside their cells is a glass, initially upside-down.
  • Each minute, the sultan randomly selects one man to visit the glass.
  • That man may flip the glass (or not), based solely on his memory.
  • Once separated, the men have no communication, and the only observable shared object is the glass.

At any point, one man may declare that everyone has been called.
If correct: all are freed.
If wrong: they all die.

They get one chance before being separated to agree on a strategy.

Goal

Design a guaranteed strategy to eventually make the correct declaration—regardless of the randomness in the sultan’s choices.

Key Idea: Designate a Counter

The strategy uses one designated counter, say Wise Man #1, and the other 49 act as signalers.

Pre-agreed roles:

  • One counter (Wise Man #1)
  • 49 signalers (all others)

The Protocol

  • The counter is responsible for counting unique signalers.
  • The glass has two states: up and down.
  • All signalers know the initial state is down.

Rules for signalers (non-counters):

  • Exactly once, when a signaler sees the glass down, they flip it up—this is their signal.
  • After doing so once, they never flip the glass again.

Rules for the counter:

  • When the counter sees the glass up, he:
    • Flips it back down.
    • Increments his internal count by 1.
  • When the glass is down, he does nothing.

Eventually, once the counter has seen 49 flips up, he knows all 49 signalers have been summoned at least once.

At that point, he declares that all 50 have been called (including himself, since he’s being called now).

Why It Works

  • Each signaler flips the glass once, the first time they’re called and it’s down.
  • Only the counter resets the glass to down and increments the tally.
  • This guarantees a one-to-one correspondence between signaler appearances and the counter’s count.
  • No false positives occur, because no one flips the glass more than once.

Eventually, the counter will tally 49 ups, proving all others have appeared.

Final Answer

Designate one man as counter.
All others flip the glass from down to up only once, the first time they’re called and the glass is down.
The counter flips it down to up, and increments a count.
When he reaches 49, he declares that all have been summoned.

Reference