# sushi disk distance problem

so I was reading through the blogs and find an interesting problem sushi disk problem. It was written in Chinese, so I am going to translate it. Correct me where I mistake something if you know Chinese

The author was having sushi with his friend in the Restaurant where sushi is served on rotating tables. After his friend had enough sushi, he began to get annoyed by the disks because they are not distributed evenly, so he pick the middle disk of the 3 disks next to each other and adjust its position to make it the center of other two, and he did it again and again. and suddenly he had this assertion: Assume no one else is touching the sushi table. if he keep on doing that, eventually all the disks will become evenly distributed.

It was a quite interesting problem to me, and he eventually proved this assertion by using some linear algebras householder transformation etc.... Well, that's too much kill in my opinion. so is there any simple proof on this ?

• my own thoughts: 3 sushi true. 4 shushi , when n-1 holds how to prove it holds for n sushi as well. Mar 2, 2014 at 14:12

I came up with this metaphor: think of the n segments between the sushi disks as n glass of hot water each with different temperature. so the corresponding operation is like take two glasses next to each other and mix them together. eventually all glasses will reach the same temperature if we don't consider energy loss. Anyways this is not formal proof.

Reductio ad absurdum Assume that after all the infinite operations applied on the sushi disks , they are still not evenly distributed. say disk A B and C. The contradiction: When we operate on Disk B we shall already make $d_{AB}=d_{BC}$. so the Assumption don't hold.

does this proof actually hold? I am not that confident actually.