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        riddles >> easy >> Chameleon island
        
(Message started by: BNC on Mar 30^th, 2003, 1:17pm)

Title: Chameleon island
Post by BNC on Mar 30^th, 2003, 1:17pm

On a remote island live chameleons in 3 colors: blue, red, and green. At a certain time, there are 13 blue chameleons, 15 red, and 17 green. No chameleon ever dies, and non born (or migrate in /out of the island).

Every time two chameleons of different colors meet, they change colors to the third one (e.g., if a red chameleon meets a blue chameleon, both turn green).

Is it possible for all chameleons to turn to the same color?

Title: Re: Chameleon island
Post by Icarus on Mar 30^th, 2003, 8:20pm

The answer is "no". Why? Because the answer to problems stated like this one is always "no". If it were possible, the problem would be stated differently! ;)

Title: Re: Chameleon island
Post by BNC on Mar 30^th, 2003, 11:19pm

You should know by now not to trust the way I word the riddles, hey? :P :-[

OK, add a last line:

If possible, show the shortest "path". If not, explain.

Title: Re: Chameleon island
Post by LZJ on Mar 30^th, 2003, 11:21pm

It's not possible. Very briefly, one has to get either 2 of the colours to have the same number of chameleons, or with the difference in number between 2 species being a multiple of 3.

(Please pardon my poor use of English)

Title: Re: Chameleon island
Post by wowbagger on Mar 31^st, 2003, 5:44am

More mathematically speaking, the three differences d_i between the numbers of chameleons of different colours aren't altered by the colour-changing when taken modulo 3.
In the beginning, all d_i = 1 (mod 3). (Or 2, depending on which you subtract. The important thing is that d_i != 0 (mod 3).)
If there's only one colour left, the difference between the other two is d_j = 0. However, this value cannot be attained by the process.
quod erat demonstrandum.