Question 77
Initially, there are $n$ glasses numbered $1, 2, \ldots, n$, and the capacity of glass number $i$ is $i$ liters. A total of $n$ liters of water is distributed among these glasses such that the volume of water in each glass is an integer. In each step, we can pour water from one glass into another until the source glass is empty or the target glass is full.
a. Prove that from any configuration (where the volume of water in each glass is an integer), it is possible to reach a configuration in which each glass contains exactly $1$ liter of water, in at most $\frac{4n}{3}$ steps.
b. Prove that from any configuration (where the volume of water in each glass is an integer), it is possible to reach any other configuration with the same property in at most $\frac{5n}{3}$ steps.
a. Prove that from any configuration (where the volume of water in each glass is an integer), it is possible to reach a configuration in which each glass contains exactly $1$ liter of water, in at most $\frac{4n}{3}$ steps.
b. Prove that from any configuration (where the volume of water in each glass is an integer), it is possible to reach any other configuration with the same property in at most $\frac{5n}{3}$ steps.