## Thursday, January 21, 2010

### Three Marked Women

While a red mark was placed on the forehead of each of three blindfolded women seated facing each other in a circle, they were told that the the mark might be either red or white. Upon removal of the blindfolds, each was to raise her hand if she saw at least one red mark, and then to take it down if she could logically deduce the color of her own mark.

All three hands were quickly raised, but then one of them lowered her hand. How did she know?

1. This comment has been removed by the author.

2. I think the only way our heroine could know would be through an educated guess. All other options lead to someone knowing for sure.

Any two have Red and other has white:
Two ladies see one red, raise hands.
One sees two red raises hand.
Two ladies who see one red deduce theirs must be Red since the only other mark they see is white. Both ladies who see white raise hands.
**This isn't the correct option.

Any two have White and the other has Red:
Two ladies see one Red. Each raises hand.
Both of the ladies who raise hands assume they have white since the third saw no red.
**This also isn't the correct option.

All three have red.
Our heroine assumes that if the scenario was W-W-R, two would be able to deduce their color and lower hand, but none do. If the scenario was W-R-R, again two of them would know they had white. Since none was quick to lower her hand, the heroine assumes all have red, none can deduce what they have, and she guesses she has Red as well.

3. The answer would be white-red-red.

The girl would see that the person across from her has red, and the other girl has white. Since the person with white on her forehead raised her hand, it means that she had to have seen the girl with the red mark. Since the girl with the red mark saw a red mark, then it means that the only person she would have seen with a red mark is our heroine.

The girl can see red and white, and since they all raised their hands, she can say without a doubt that she has red.

4. Alex,

I'm not sure how understandable my response was, oh well.

But if it was W-R-R, then wouldn't two women be able to deduce their color?

Woman with white: Sees two red, raises hand, leaves hand up, has no idea what color she has.

Both women with red: See one red, one white, they each raise their hand. They both deduce, that since the other person sees only white and some other color, and if that color is red, they raise their hand, and they did raise their hand, each deduces that they must have red in order for the other woman to have raised their hand. Each Red woman knows the other red woman sees white and red and each red woman lowers her hand knowing she has red.

The puzzle states only one lowered her hand not two, and with W-R-R, I think two could lower their hands.

5. This is unlike some of the questions I have posed here, because we are not assuming everyone in the puzzle has perfectly logical reasoning. So, while one of the women in the puzzle has figured out the solution, any of the three could have (and if we continued our story, probably did in a few more moments).

Back to our heroine: Her first thought is, I have either red or I have a white spot. Suppose I do have a white spot. Then the other two women would be looking at one white spot and one red spot. They would quickly figure out the other woman had her hand up because of their own red spot.

Since neither of them figured it out, I must have a red spot.

6. Mike, good reasoning... but I was thinking that she had better eyesight, and saw the reflection of her own dot on the others' eyes. Otherwise the same logic applies to all of them: there must be some differentiating factor.

7. Yeah, but if they all have red, they all would have seen eachother with their hands up, and they all would have put their hands down at the same time. So this one really isn't solve-able unless you know that one person is smarter then the others. I don't think that there's really any case where only one person would be able to figure out their color. Plus in the situation of all have red, the one who put her hand down first could have thought that they had white, because they saw two red marks and thought that the test giver meant that one person had to have white.

Since the solution has to assume one person is smarter then the others, this indicates that either W-R-R, or R-R-R would be correct, and the first person to lower their hand would assume they had red on their forehead, even if they had white.

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