In a distant dictatorship, where everyone is perfectly rational, a new disease appears.
Its only symptom is unmistakable: an infected person’s shadow becomes twisted and distorted. For some mysterious reason, however, people cannot perceive the change in their own shadow, even though everyone else can see it clearly.
News travels fast. Everyone knows which other people are infected, but no one knows whether they themselves are infected. As the disease spreads, the tyrant proclaims two new laws:
- “To avoid unnecessary panic, it is strictly forbidden, under penalty of death, to reveal another person’s condition to them.”
- “Anyone who becomes certain that they are infected must leave the country permanently before sunrise.”
Every morning, the newspaper publishes the number of people who left the country during the night.
For 22 days after the laws are proclaimed, the newspaper reports that no one has left.
But on the 23rd morning, several people are gone.
How many?
Hints
1Hint 1
Open this only if you want a small nudge before looking at the solutions.
Solutions
1Reveal solutionsAre you sure? Give it a try first.
Solution by AndurilFR
Procédons par récurrence sur le nombre d’infectés pour montrer que N infectés partent au bout de N jours.
Initialisation : S’il y a un unique infecté, il sair que la maladie existe mais ne voit aucun malade. C’est donc le seul. Il part immédiatement.
Hérédité : Supposons qu’il y a N+1 malades. Un infecté voit N malades seulement. Il y a deux cas : soit il n’est pas infecté et il y a exactement N malades qui partiront donc au N eme jour, soit il est infecté et il y en a N+1. Il doit attendre le N ème jour pour le savoir. Le N ème jour, personne ne part car, étant parfaitement rationnels, ils ont tous le même raisonnement.
Ainsi, ils comprennent qu’il n’y a pas N infectés mais N+1 et qu’ils sont donc infectés et partent tous le N+1 ème jour.
Conclusion : 23 personnes partent.