When one move on a ring randomly, the expected time to coming back to the origin is equal to the perimeter according to this.
At frist, the man is at the origin. For each second, he move left or right by 1 meter randomly. The perimeter of the ring is $n$ meter. Then the expected time of coming back to the origin is $n$ seconds which can be obtained by solving reccurence formula.
I want to know the interpretation (or feeling) of this result since the result is very simple. Is there any other simple way to get that the expected time is euqal to the perimeter.