Three Aztecs are dressing for a football game. They know there are 5 helmets in the locker, 3 black and 2 red. As they are about
to get their helmets, the lights go out. They all decide to just grab a helmet in the dark and head for the field. As they go in single file out into the tunnel, the one in the back says, "I can see both of the helmets in front of me, but I do not know the color of my helmet". The one in the middle says, "I can see the helmet in front of me, but I cannot determine the color of my own". The one leading the pack says, "I can see neither of the helmets behind me but I know the color of my own". How does he do it and what color was the helmet?
Let us start from the guy at the back. The guy at the back
can see the two helmets in front of him. If the two helmets in front of him are
red, then he can certainly say that his helmet is black since there are only 2
red helmets. Since that is not the case, then therefore he can either be black
or red. Therefore this also means that at least 1 of the 2 guys in front of him
has black helmet.
<span>Since the remaining two should have at least 1 black,
therefore the middle guy can see that the helmet in front of him is not red (so
1st guy is black). Because if the guy in front of him is red, then
he can know that he must be black since remember, at least one of the 2 front
guys must be black, but the middle guy cannot say that. Therefore the 1st
guy knows he has a black helmet.</span>
Find the prime factorization of 54 54 = 2 × 3 × 3 × 3 Find the prime factorization of 99 99 = 3 × 3 × 11 To find the GCF, multiply all the prime factors common to both numbers:
