# Is it possible (for some configuration of initial 9 flowers) to get all red flowers after finitely many years?

An isolated garden has the shape of a circle. Initially, there are 9 flowers on the circumference of the garden: 5 of the flowers are red and the other 4 are yellow. During the summer, 9 new flowers grow on the circumference of the island according to the following rule: between 2 old flowers of the same color, a new red flower will grow, and between 2 old flowers of different colors, a new yellow flower will grow. During the winter, the old flowers die, and the new survive. The same phenomenon repeats every year.

Is it possible (for some configuration of initial 9 flowers) to get all red flowers after finitely many years?

I do not think this is possible because in order to get all of the flowers to become red you first need to get all of the flowers to become yellow. In order for all of the flowers to become yellow we will need the colors of the flowers to alternate and we will need an even number of flowers for these alternating flowers to make all of the new flowers yellow, therefore it is impossible.

Is it impossible and is my reasoning correct?

• BTW, I don't think MathJax is necessary when it is just numbers. It doesn't look much better, it slows rendering and it makes the title look funny on the home page. – boboquack Nov 29 '16 at 1:27

It is impossible, and your reasoning is correct.

Slightly more rigorously, for all the flowers to be red:

Either previously all the flowers were red, or all the flowers were yellow.

So at some point:

All the flowers were yellow.

Then:

All the flowers alternate back and forth. So there must be a red flower.

Then:

Going around the circle we have red-yellow-red-yellow-red-yellow-red-yellow-red.

But then:

We have two red flowers next to each other, and a red flower will grow between them.

So:

The circle can never be completely yellow, and therefore never completely red.

Note:

This proves that it is not possible from ANY starting position that is not monochrome.