Logic problem

Kevin · 2003-07-14, 04:16

Not very hard, but it's the first decent one I found in my"Mathematical Thinking" book.

Three children are in a line. From a collection of 2 red hats and 3 black hats, the teacher places a hat on each child's head. The third child sees the hat on the first two, the the second child sees the hat on the first, and the first child sees no hats. The children, who reason carefully (and for clarity, perfectly) are told to speak out as soon as they can determine the color of the hat they're wearing. After 30 seconds, the front child correctly names the color of her hat. What color is it, and why?

eepiccolo · 2003-07-14, 15:12

Hey, I actually figured this one out on my own! My brain still works; I thought it had permanently shut-down after college ;)

The first child is wearing a black hat, and she knows this because the other two "told" her, but without them saying anything.

This is how it goes, and I will refer to the first, second, and third child as One, Two, and Three, respectively:

Three would know that he was wearing a black hat if the others were both wearing red because there are only two red hats. But Three was silent, so Three didn't know what he was wearing, so three possible hat configurations exist for One and Two, One has red and Two has black, or One has black and Two has red, they both wear black.

Two realizes that only these configurations exist, so he looks to see if One has a red hat, for that would mean that Two would have a black hat. But Two was silent, which meant One was wearing black.

One realizes what both Two and Three have determined, and because of their silence, One knows she is wearing a black hat, and speaks up.

rpresser · 2003-07-15, 21:21

I guess there's still no way that either One or Two can know what Two is wearing, or that anyone can know what Three is wearing?

S80780 · 2003-07-16, 05:48

That's true. The only combination allowing each child to determine the color of its hat is One and Two wearing red hats and Three wearing a black hat. This way Three could state to wear a black hat and with this information, both One and Two would know that their hats are both red.
If One wears a red hat and Two a black one, Two would first state to wear a black one and then One would state to wear a red one. But Three could not determine the color of its hat which is black with a probability of two thirds.
In the given situation, Two has a fair chance (50%) to guess its hat's color and Three has a chance of one third to wear a hat of the same color as Two.

Benjamin

2003-07-14, 04:16	#1
Kevin Aug 2002 Ann Arbor, MI 433₁₀ Posts	Logic problem Not very hard, but it's the first decent one I found in my"Mathematical Thinking" book. Three children are in a line. From a collection of 2 red hats and 3 black hats, the teacher places a hat on each child's head. The third child sees the hat on the first two, the the second child sees the hat on the first, and the first child sees no hats. The children, who reason carefully (and for clarity, perfectly) are told to speak out as soon as they can determine the color of the hat they're wearing. After 30 seconds, the front child correctly names the color of her hat. What color is it, and why?

2003-07-15, 21:21	#3
rpresser Jul 2003 A₁₆ Posts	But then ... I guess there's still no way that either One or Two can know what Two is wearing, or that anyone can know what Three is wearing?

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2003-07-14, 15:12	#2
eepiccolo Dec 2002 Frederick County, MD 172₁₆ Posts	Hey, I actually figured this one out on my own! My brain still works; I thought it had permanently shut-down after college ;) The first child is wearing a black hat, and she knows this because the other two "told" her, but without them saying anything. This is how it goes, and I will refer to the first, second, and third child as One, Two, and Three, respectively: Three would know that he was wearing a black hat if the others were both wearing red because there are only two red hats. But Three was silent, so Three didn't know what he was wearing, so three possible hat configurations exist for One and Two, One has red and Two has black, or One has black and Two has red, they both wear black. Two realizes that only these configurations exist, so he looks to see if One has a red hat, for that would mean that Two would have a black hat. But Two was silent, which meant One was wearing black. One realizes what both Two and Three have determined, and because of their silence, One knows she is wearing a black hat, and speaks up.

2003-07-16, 05:48	#4
S80780 Jan 2003 far from M40 5³ Posts	That's true. The only combination allowing each child to determine the color of its hat is One and Two wearing red hats and Three wearing a black hat. This way Three could state to wear a black hat and with this information, both One and Two would know that their hats are both red. If One wears a red hat and Two a black one, Two would first state to wear a black one and then One would state to wear a red one. But Three could not determine the color of its hat which is black with a probability of two thirds. In the given situation, Two has a fair chance (50%) to guess its hat's color and Three has a chance of one third to wear a hat of the same color as Two. Benjamin