Arthur and the Knights of the Round Table

Problem:

King Arthur and nine knights are seated around the Round Table. The king has a pile of ten plates in front of him and the knights do not have any plates. In one step, a person having two plates is authorized to give one of his plates to his left neighbor and, at the same time, another of his plates to his right neighbor. Can we get to a situation where everyone has exactly one plate?

Solution

It is not possible to distribute the plates so that each knight has one plate in front of him. Indeed, separate the knights into two groups as shown in the picture. Note that every time a knight of the white group distributes two plates, two plates arrive in the hands of a knight of the black group. In other words, during each movement, an even number of plates is transferred from a group to another group. However, each group consists of five knights. Since King Arthur has 10 plates (an even number of plates) and since 5 is not even, it is impossible to distribute the plates so that each knight has one plate in front of him.