## Friday, March 7, 2014

### Four Women and Four Hats Puzzle

Once upon a time four women bought four hats. All the hats were black inside. Two of them were while on top and two black on top as in the picture below. In this game, three women (Beth, Iris, Michelle) were asked to stand inside the room facing each other's back, and the fourth (Katy) was left behind the wall.  Everyone was allowed to look only forward. Michelle saw Beth and Iris'es heads. Iris only Beth's head. Beth and Katy were facing the non-reflective wall. Then the hats were placed on their heads.

No woman knew what color hat she has, only that there are two white and two black hat overall. They were asked to figure out their hat color. No guessing was allowed, just pure logic and definite answer. They were all very savvy. After a minute of silence one of them announced her hat color. And she was absolutely right. Who was it and how did she figure it out?

This puzzle came from my daughter's friend (10-years old) and they had great time enjoying it at school. You can do it as well!

Your thoughts and suggestions are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon

SteveGoodman18 said...

Iris figured out her own hat color.

Michelle would know right away what her hat color was if she saw Beth and Iris wearing the same color. Since she didn't say anything for a minute, Iris trusted that Michelle saw two different colors. Knowing that she and Beth were wearing different colors, Iris simply announced that she was wearing the color that Beth wasn't.

Katrina Collins said...

Iris answered. Since Michelle didn't answer, she must not be able to see two white hats or she would have known she had a black hat. Therefore Iris can be sure she had a black hat.

Iris is the one who got the correct answer. She didn't answer immediately, but when Michelle didn't say anything Iris knew she must have a different colour hat to Beth; if Iris and Beth both had black hats, Michelle would have said "white" straight away; similarly if they were both wearing white hats Michelle would have said "black". After leaving Michelle time to answer and hearing only silence, Iris correctly deduced that her hat must be the opposite colour to Beth's; seeing Beth's hat to be white, she therefore said "black".

Dennis (of Dennis and Katrina) said...

I love these logic/induction puzzles!

Logically, Katy and Beth should remain silent. They can't see anyone else's hat (and can't tell their own color), so they should never speak.

Michelle is interesting - if she could see two of the same color hats, she would know that her hat MUST be the other color. One can assume that she would recognize this fact fairly quickly. Since in the puzzle, she's silent for a minute, she must be seeing one of each color.

That leaves Iris. Since Michele has been quiet for one minute, Iris knows she's seeing one of each color hat. Since Iris sees a white hat on Beth, her hat must be black. So Iris speaks up and says her hat is black.

Dennis

Anonymous said...

TracyZ:
Not sure, but I think the answer is related to the type of wall between Katy and Beth.

Option 1: the wall could be partially made of clear glass but then coated so as to make it much easier for Katy to see through the wall than for the other three women to do so. In this case, Katy is able to correctly and without guessing state her hat color.

Option 2: Another option here could be that the wall is transparent but the room with the three women is well lit and the Katy is standing in the dark (or a much darker room) and therefore can see into the room better than the women can see out. (similarly to a house at night where passers by can see in the windows, and the house's residents not see the people looking in.) In this case, Katy is able to correctly and without guessing state her hat color.

Option 3: The wall between Katy and Beth is not a full floor to ceiling wall, but a shorter dividing wall. Only one of Katy or Beth is tall enough to see over the wall. That person then will be able to correctly and without guessing state her hat color.

An aside (Option 4?): I am still wondering a bit why these women bought hats without knowing the color of them, and how they then put them on their heads without seeing what color they were. If one of the four woman did the hat buying and then distributed the hats to the other three women, she should be able to know what hat each of them, including herself, is wearing, and state as much without any guessing.

Annie said...

Iris would be the one to answer. A minute passed with no answer so Iris can be quite sure that she does not have the same color hat as Beth. If they had the same color it would be easy for Michelle to guess her color hat which would be the opposite of Iris and Beth's hats. Since Michelle did not answer Iris can be sure that she and Beth have different color hats so she answers that she has the opposite color of what she sees Beth wearing!

tom said...

Iris it is. Iris can deduce logically that since Michelle has not spoken, Michelle must be seeing a mix (one of each). Since Iris sees a white hat, hers must be black.

The puzzle, and this answer, assumes Iris knows that Michelle is correct in her silence.

Jerome said...

I am guessing that Iris shouted out that her hat was black.

Michelle sees 2 hats, one of each color, but she says nothing. That means that the hats are not the same color. There is one of each, and Michelle cannot say with certainty which kind she has on her head. The fact that Katy is effectively in another room does not help Michelle at all. All Michelle can gather from that is that Katy is wearing a hat.

From Michelle's point of view, she herself could be wearing either.

Now Iris puts on her thinking hat and says to herself "Michelle is silent. She does not know what color she wears. That means she cannot tell from what she sees. Since I see a white hat on Beth, I must have on a black hat."

Anonymous said...

Had Iris and Beth been wearing the same color hat, Michelle would have instantly figured out her hat color. However this didn't happen and so we would assume that Iris and Beth weren't wearing the same color hat. Since Iris could see Beth's hat color, she could easily deduce that she was wearing a different color.
My guess would be Iris.

Lulu

Maria said...

This is a fun puzzle and I see you enjoyed untangling it.
Katrina - it is so nice to see you participating and ahead of Dennis (who is her husband).
TracyZ - what happened? You must have been thinking Illusionists instead of Agatha Christie.
SteveG., Katrina, AdamT, Dennis, Annie, Tom, Jerome, Lulu - everyone who solved correctly get a puzzle point. Share this one with your kids.