There are

cups labeled
, where the
-th cup has capacity

liters. In total, there are

liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most

steps.

Prove that in at most

steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
This post has been edited 1 time. Last edited by M11100111001Y1R, May 27, 2025, 7:26 AM