TITLE: chameleon puzzle- modified one
QUESTION [8 upvotes]: In one island there are 3 colors of chameleon 12 blue,15 green and 7 red. When two different color’s chameleon meet together , they convert into third color. What is the number of minimum no. of meeting required to convert all chameleon into same color?

REPLY [3 votes]: You can work backwards from a sea of a single color (say, blue), counting the number of "unmeetings" required.  A single (production) unmeeting is required to produce one of each non-sea color (say, $2B\rightarrow G+R$), or a single (conversion) unmeeting can convert two of one non-sea color to one of the other (say, $2G\rightarrow R+B$).  If two conversion unmeetings have different colors, then they have the net effect of reducing each non-sea color by one; but this can be done more cheaply by just reducing the number of production unmeetings by one.  So all conversion unmeetings will be of a single color.  The result will have $n_1=x-2y$ chameleons of the first non-sea color, and $n_2=x+y$ of the second non-sea color, so $n_2-n_1$ must be a multiple of three, and the number of unmeetings will be $x+y=n_2=\max(n_1,n_2)$.  In other words, look for two colors whose counts differ by a multiple of three; the result is the larger of the two counts.
In your case, the two colors whose counts differ by a multiple of three are green ($15$) and blue ($12$); to produce the observed counts from a sea of red takes $15$ unmeetings, consisting of $1$ conversion unmeeting and $14$ production unmeetings.  In the other direction, that is, first $R+G\rightarrow 2B$, and then $14B+14G\rightarrow 28R$.
Note that there are configurations with no solution.  For instance, given $10$ red chameleons and $2$ blue chameleons, there is no way for all chameleons to reach a single color.  One the other hand, starting from $6$ red chameleons and $3$ blue chameleons, it is possible to achieve either $9$ red chameleons (with $3$ meetings), or $9$ green chameleons (with $6$ meetings), or $9$ blue chameleons (with $6$ meetings).