数値円陣

Let's say we have five #1's, and four #0, arranged randomly in a circle. Between every two numbers that are the same, you put a 1, and between every two numbers that are different, you put a zero. After doing so, you remove the previous set of numbers.

The question now is: Is it ever possible to have a circle of nine #0's?

Let's look at a random distribution first:

I use Mahjong tiles that are face up to denote #0, and face down tiles to denote #1. The inner ring is our random distribution of numbers in a circle, while the outer ring is the result of performing the addition processes mentioned above. From our test of the random distribution, we can see that in order for any chance that there could be nine #0's, we should arrange the numbers to alternate in a circle as so.

But a problem arises, which we can see if we repeat the process.

Our result is eight #0's, and one #1.

In conclusion, no, there is no distribution of the given numbers that will result in a sequence of nine #0's.

Since the total numbers in the circle are odd, and the fact that we have one more #0 than we have #1, the circle is bound to have at least one pair of repeating numbers (since the first number connects with the last number to form the circle). No matter how many times we repeat the procedure, the maximum number of #0's will be eight.