Long ago, there existed a species of fighting
chameleons. These chameleons were divided into six types of
matching color and strength:
Black were the strongest, followed by
blue,
green,
orange,
yellow and
white which were the weakest.
Whenever two chameleons of the same color met, they would fight to the
death and the victor would become stronger and change color (eg white to yellow). Black chameleons would fight eternally.
The small island of Ula was initially populated by a
group of fighting chameleons. For this group
a) the colors present each had an equal number of
chameleons (for example, group = 3 black, 3 green and 3 yellow)
b) it was not made up entirely of white chameleons
After all the possible fighting was done, there remained one black and
green and no blue or orange chameleons.
How many white chameleons remained in the island?
Prove it.
The original fighting chameleon population would have been one of the following:
Black Blue Green Orange Yellow White
A) 1 0 1 0 0 0
B) 1 0 1 0 0 1
C) 1 0 1 0 1 0
D) 1 0 1 0 1 1
E) 0 2 0 2 0 0
F) 0 2 0 2 0 2
G) 0 0 3 3 3 0
H) 0 0 4 0 4 0
I) 0 0 5 0 0 0
J) 0 0 0 6 6 6
K) 0 0 0 8 0 8
L) 0 0 0 10 0 0
M) 0 0 0 0 14 14
N) 0 0 0 0 20 0
For cases A thru D no combat could occur as there are not two or more chameleons of the same color. For cases E thru N, where combat can occur, there are only four cases where there are any white chameleons. As there are an even number of white chamelons, after each pair engaged in combat, all the victors would change from white to yellow leaving no white chamelons. Therefore, unless the beginning population began with only with 1 white chamelon, no white chameleons would remain on the island after all possible engagements.
Edited on May 4, 2007, 5:30 am
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Posted by Dej Mar
on 2007-05-03 23:51:30 |
Solution
