mersenneforum.org this is a toughie
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 2006-11-17, 23:27 #1 geminiasq   146568 Posts this is a toughie Hey all, I need some help solving this problem...I dont even know how to begin it. The bio geek in me is saying to do a ton of punnet squares but the math in me knows theres a better way to find the answer....maybe a pattern ok so heres the problem. The country of Bunnylandis inhabited by red, blue, and green rabbits. When two different color rabbits meet they both turn into the third color. Example: if a blue and a green rabbit meet, they both become red, etc. On November 4, 2006 at 10:00 pm the population of Bunnylandconsisted of 363 red, 854 blue, and 220 green rabbits. Can all the rabbits of Bunnylandturn into the same color? If your answer is yes, you should present an example (sequence of rabbit meetings) that results in turning all the rabbits intothe same color. If your answer is no, you should present a proof.
 2006-11-18, 07:58 #2 S485122     Sep 2006 Brussels, Belgium 2·7·113 Posts Go backwards : start with all rabbits of one color L. You will see that the difference in the numbers from the two other colors O1 and O2 will always be a multiple of three : You start with a difference between O1 and O2 of 0 a multiple of three. If you take a number n off a color, it means that a meeting has just happened between an equal number of rabbits of the two other colors. n = 2m What happens during meetings, there are two cases : - A meeting took place between a number m of rabbits of the the two "other colors". We can compute the preceding situation. L -> L - 2m O1 -> O1 + m O2 -> O2 + m The difference between the number of rabbits of the "other" colors does not change. - A meeting took place between one of the other color and the remaining color. We can compute the preceding situation. L -> L + m O1 -> O1 + m O2 -> O2 - 2m The difference between the number f rabbits of the "other" colors has changed by 3m. In other words the difference will allways remain a multiple of three. Since in the initial conditions no difference between numbers is a multiple of three, you cannot arrive at a monocolored population.
 2006-11-18, 08:36 #3 Citrix     Jun 2003 32×52×7 Posts The answer is simple. Yes they can all converge into the same color, assuming the initial population of rabits all dies off. If only 1 color rabits breed then only 1 color rabbits remain. If you want a better answer, describe what parameters control the population of the rabbits.
 2006-11-18, 08:55 #4 akruppa     "Nancy" Aug 2002 Alexandria 46408 Posts As I understood it, these rabbits are both immortal and infertile. Rather atypical of rabbits, imho, but a nice puzzle nonetheless. Alex