Question 102476: three finalists in a beauty pagent are placed in line waiting to on stage. (they are lined up single file, like you do in kindergarten.) they are told to close there eyes and three tiaras are taken from a bin containing 3 silver tiaras and two gold tiaras, and one tiara is placed on each of there heads. they are then told to open there eyes. the last woman in line, (who can see the other two), says i wonder what color of tiara i am wearing? the middle women (who can only see the woman in front of her me to i wonder what color i am wearing.) the first woman (who can see neither of the other woman) says i know what color my tiara is.
what color of tiara is the first woman in line wearing. write a short paragraph how she determined the color of her tiara.
my question was not from my book
Answer by mjgulinello(10)   (Show Source):
You can put this solution on YOUR website! For this problem to work, you have to assume that the beauty pageant contestants would be able to solve this problem themselves.
Anyway, here is the solution.
First of all, it doesn't matter which gold tiara is which or which silver tiara is which. The only important thing is whether the tiara is gold (G) or silver (S).
Therefore we have a combinations problem.
There are 7 possible combinations for the tiaras (back - front)
SSS
SSG
SGS
GSS
SGG
GSG
GGS
There is one impossible combination: GGG, because there are only 2 gold tiaras available.
Our first fact is that the woman in back did not know what color tiara she had on. Even though she could see the two women in front of her, she still could not tell. What we want to do with this information is try to eliminate some of the possible cases. To do this we need to figure out if there are any cases for which the woman in back would have known the color of her tiara.
It turns out there is one.
If she had seen two gold tiaras in front of her, she would have known hers was silver. This is because there is only one valid case with _GG: SGG. Since GGG is not a valid case, she would have known her tiara was silver.
Since she did not know, we can eliminate SGG from the list.
Now we are left with 6 cases.
SSS
SSG
SGS
GSS
GSG
GGS
The woman in the middle now starts with the information that the woman in back did not see two gold tiaras. So if she saw a gold tiara in front of her she would know hers must be silver. Since she did not know, then she must have not have seen a gold tiara in front of her. This eliminates the cases of SSG and GSG.
We are left with 4 possible cases.
SSS
SGS
GSS
GGS
Now the woman in front knows that no matter which of these cases is true, her tiara must be silver, as it is in all 4 remaining cases.
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