Freedom! Also, chameleons.
Posted by Rob Herman at April 2nd, 2007
Tonight, I cancelled my World of Warcraft account. I look forward to having more time to read, work out, blog, design games, hang out with friends, play other computer games, swing dance, cook, go to concerts, watch the Daily Show, and more. The extra bonus is that I can probably get all of these activities and more into the newly-freed hours.
To inaugurate the new era, I present to you this riddle: An island is populated with chameleons in three colors: red, green, and blue. When two chameleons of different colors meet, they both became the third color. So if a green and a blue chameleon meet, they both become red.
At one time, there are 13 red chameleons, 15 green chameleons, and 17 blue. Is it possible for all the chameleons to ever be the same color? Why or why not?
Solution alert.
Clearly, in order for all chameleons to become the same colour, the number of chameleons of two other colours must become equal at some n > 0.
The differences between the numbers of each colour are 2, 2 and 4. We have three similar operations that work upon the differences. These operations can:
a) add 2 to one of the colours and subtract 1 from the other
b) as per a, inverting the two colours
c) subtract 1 from each colour, not changing the difference in the numbers of chameleons
Therefore, we can only change the difference between the number of chameleons in two difference colours by multiples of 3. Since the differences are not multiples of 3, none of them can be reduced to 0. Thus it is not possible for all of the chameleons to ever be the same colour.
That works.
This one took me longer than it really should have to work out. What I eventually came up with was this:
For all of the chameleons to be the same color, the numbers of two colors of chameleons must be the same mod 3 (as it turns out, they must both be 0). But the only allowed operation reduces the number of each color of chameleons by exactly 1 mod 3. Since the values all start out as different mod 3, there is no way that all the chameleons can ever be the same color.