Question 12
Elisa has 2023 treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules:
- if more than one chests are unlocked, it locks one of them, or
- if there is only one unlocked chest, it unlocks all the chests.
Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to lock.
- if more than one chests are unlocked, it locks one of them, or
- if there is only one unlocked chest, it unlocks all the chests.
Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to lock.