Question : You have a rose garden. You have planted 16 rose plants in a circle. Some of them are red and some are white. These plants are somwhat strange. They flower according to the following rules:
1. Each plant has a new flower everyday.
2. Each plant bears only one flower on any day.
3. The color of the flower on a plant depends upon the colors of its neighbours’ flowers on previous day. If both the neighbours had same color on previous day then the plant has red flower and if the neighbours had different colors then the plant has white rose.
After many days you spot that all the roses have become red. But you want both colors to live. So you start with a different initial arrangement. But to your disappointment, you again find that after sufficient number of days, all the flowers have turned red. You are smart enough to deduce that given any initial arrangement, you always end up with all red flowers. What I want to see is whether you are smart enough to find out why this happens and for which numbers other than 16, the same thing happens.
Answer : If the number of plants in the circle is (2^n) where n is any integer then all flowers turn red after sufficient number of days irrespective of the initial arrangement.
Its difficult to describe the full method here. But I am giving you pointers which will easily lead to the solution.
You can assign 0 to white and 1 to red.
First you can show that principle of superposition is valid. For example, you can split a case with k red flowers into k seperate cases each having only one red flower at appropriate position and analyse. After going forward by say, p number of days, you can again superpose the k cases so that you get the original thing.
This makes the problem easy. Now we can analyse the “one flower red” case in detail.
After analysing this case in detail, you will observe the following.
After (2^k) days, only (2^k)th flowers on either of the sides of the original red flower are red. So if the (2^k)th positions on left and right coincide, then by superposition, all the flowers will become white.
This shows that the number of flowers has to be (2^k)
Its not so straightforward. Put a little thinking into this and you will get what I want to say.