50 Wise Men & the Glass
A coordination puzzle under uncertainty. One man is chosen as the counter; the other 49 each contribute exactly one signal. Random selection changes only how long the process takes — not whether it is correct.
Glass: UPSIDE-DOWN · ready to receive a signal
0 unique signals countedSafe declaration at 49
They agree beforehand that Wise Man #1 will be the counter.
The strategy
1
Choose one counter. Wise Man #1 never contributes his own signal.
2
Each of the other 49 men remembers one bit: “Have I signaled yet?”
3
If a non-counter who has never signaled sees the glass upside-down, he flips it right-side-up. That is his one signal.
4
Whenever the counter sees it right-side-up, he flips it back and increments his count.
✓
When his count reaches 49, all 49 other men must have signaled, so all 50 have been called.
Why this can never give a false declaration
A count is added only after a non-counter creates a right-side-up signal, and every non-counter is allowed to create that signal at most once. Therefore 49 counts correspond to 49 distinct men.
Random selection affects only how long it takes, not correctness. With independent random calls, every man is eventually selected with probability 1.