Which Mark | 2 Sep 2009

shabbir's Avatar, Join Date: Jul 2004
Go4Expert Founder
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?

Now if you Google the above query you would find the answer. I want to know the explanation of the answer you have found.
naimish's Avatar
Banned
This is a variation of the above problem, but very hard for some people to solve. Everyone can see that each woman could be thinking that her own forehead might be red or white because the other two women could be raising their hands because of each other. Again, this is the point where most people stop, and indeed, where the women stopped while all three hands were up. The trick to solving this problem is to REALLY put yourself into the smart woman's shoes. If you REALLY were she, you'd say, either I have a white spot or a red. Suppose you had a white spot. Then the other two women would be looking at one white spot and one red. They would each quickly figure out that that only reason the others hand was up was because of their own red spot. The fact that neither of them figured it out was the tip off to the first that she must also have a red spot. Most people put themselves in the place of the first person, but to solve this one, you must then also put yourself in the place of a second woman.
naimish's Avatar
Banned
If hers was white, then one of the other 2 women with Red dots would have known their color. The way they would have known is .... each of them would be seeing 1 Red and 1 White which means they cannot be White since all 3 raised their hands, and would have guessed their color to be Red, but they did not. This way she derives that hers cannot be White, and only can be RED!
anju.hifi's Avatar
Go4Expert Member
if person A is the one who gets it correct:

A looks at B and C, both are red

he notices:
if As was white then - B would have lowered his hand knowing that C must have his hand up because of his own red spot
Mridula's Avatar, Join Date: Mar 2008
Pro contributor
For example : X, Y & Z are the women here!!

Say X has lowered her hand and found the answer. Because of below reasons :-

1. Mainly by seeing Y & Z expressions on rasing their hands and not lowering their hands.
2. Because, X would think, if at all, white spot is their in her own forehead, then Y and Z would have raised their hands, only by seeing each other's forehead's RED spot. And very moment, they could have answered their own forehead colour and thats nothing but as RED. But they did not answer.
3. It means, Y & Z have both raised their hands, one of them by seeing X's forehead as "RED".
4. But now, X can see very much RED Sppot on both Y & Z's Forehead and also see them not raising their hand, so she (X) thinks, Y & Z may not have seen each other's.
5. Since one of them have raised their hand by seeing X only, has been decided and confirmd by X now.

And so, X after a delay, smartly lowers her hand to say her forehead is colored with RED.

Hope this explains
Mridula's Avatar, Join Date: Mar 2008
Pro contributor
To add more to the above explanation:

At step-4 :

X concludes very much that Y & Z have raised their hands, only by, one of them seeing "X" 's forehead's as RED.
As in this situation, if they have seen each other's forehead only (and also if X's is White), they could have easily answered and so X is concludes his forehead colour.
Mridula's Avatar, Join Date: Mar 2008
Pro contributor
Here with a better explanation:

For Example: X, Y, Z are the women

Condition is all of them have raised their hands, means all of them have seen
atleast one person's forehead marked with "RED"

Now, lets say "X" is the person who has answered her forehead's colour.

How:

if X == WHITE

then Y & Z must be RED, to satisfy the above condition as :

X can see either Y or Z forehead colour as "RED"
Y can see only Z as X has WHITE
Z can see only Y as X has WHITE

In this scenario, very next moment, either of them i.e. Y or Z could have answered their own
forehead's colour as "RED".

But they didn't answer instead the answered person is "X", it means

Either Y or Z has raised their hand by seeing X's forehead's colour as "RED" and
not seeing each other's forehead's colour as "RED".

So, this makes X to answer that her forehead is marked with "RED" !!
shabbir's Avatar, Join Date: Jul 2004
Go4Expert Founder
Quote:
Originally Posted by naimish View Post
This is a variation of the above problem, but very hard for some people to solve. Everyone can see that each woman could be thinking that her own forehead might be red or white because the other two women could be raising their hands because of each other. Again, this is the point where most people stop, and indeed, where the women stopped while all three hands were up. The trick to solving this problem is to REALLY put yourself into the smart woman's shoes. If you REALLY were she, you'd say, either I have a white spot or a red. Suppose you had a white spot. Then the other two women would be looking at one white spot and one red. They would each quickly figure out that that only reason the others hand was up was because of their own red spot. The fact that neither of them figured it out was the tip off to the first that she must also have a red spot. Most people put themselves in the place of the first person, but to solve this one, you must then also put yourself in the place of a second woman.
All from the source.

Quote:
Originally Posted by naimish View Post
each of them would be seeing 1 Red and 1 White which means they cannot be White since all 3 raised their hands, and would have guessed their color to be Red, but they did not. This way she derives that hers cannot be White, and only can be RED!
Could not get the full logic.

Quote:
Originally Posted by anju.hifi View Post
if person A is the one who gets it correct:

A looks at B and C, both are red

he notices:
if As was white then - B would have lowered his hand knowing that C must have his hand up because of his own red spot
Could not understand much and also why RED

Quote:
Originally Posted by Mridula View Post
Here with a better explanation:

For Example: X, Y, Z are the women

Condition is all of them have raised their hands, means all of them have seen
atleast one person's forehead marked with "RED"

Now, lets say "X" is the person who has answered her forehead's colour.

How:

if X == WHITE

then Y & Z must be RED, to satisfy the above condition as :

X can see either Y or Z forehead colour as "RED"
Y can see only Z as X has WHITE
Z can see only Y as X has WHITE

In this scenario, very next moment, either of them i.e. Y or Z could have answered their own
forehead's colour as "RED".

But they didn't answer instead the answered person is "X", it means

Either Y or Z has raised their hand by seeing X's forehead's colour as "RED" and
not seeing each other's forehead's colour as "RED".

So, this makes X to answer that her forehead is marked with "RED" !!
Great.
Mridula's Avatar, Join Date: Mar 2008
Pro contributor
thanks !!
SaswatPadhi's Avatar, Join Date: May 2009
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Congrats Mridula, nice answer