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# Colored hats, revisited

## Question

BMAD, plasmid and Rocdocmac are seated in chairs and behind screens such that each can only see the heads of the other two. And, of course, not their own. bonanova blindfolds them and places a hat on each. The hats are drawn at random from a box containing (1) Red hat, (2) Yellow hats and (3) Blue hats. i.e, { R, Y, Y, B, B, B }. Blindfolds are then removed, and each of them writes and signs a truthful statement, as follows:

• BMAD: My hat is one of two colors
• plasmid: My hat is one of three colors
• Rocdocmac: My hat is one of three colors

Thalia, who is in the next room and can't see any of the hat-wearers, but who originally placed the six hats into the box, is given the written statements and now calls out the correct color of each person's hat. What are they?

Edited by bonanova
Corrected an error in the wording of the puzzle, thanks to aiemdao

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Spoiler

BMAD wears a blue hat and the other two wear yellow hats

Edited by rocdocmac
Moved to spoiler
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Spoiler

1st person has on a blue hat;  2nd and 3rd persons have on yellow hats

1)  2nd and 3rd persons cannot see a red hat or they could not state 3 poss for their own hats. Therefore each must see a yellow and a blue

2)  1st person has 2 poss ...A) he sees a red and a blue ( void by next statement );   B)  he sees two yellow

3)  if 1st saw a red ,then 2nd or 3rd would have to see a red ,but since neither 2nd or 3rd can see a red, then  poss  A) for 1st is void.

4) Therefore option 2) is correct for 1st person i.e. 1st person sees two yellow hats.

5)  Since by 1) 2nd and 3rd persons cannot see a red hat and since each have on yellow hats / option B) it then follows that 1st person has on a blue hat

Edited by bonanova
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something wrong

BMAD: My hat is one of two colors

=>  P and R 's hat is : RY, RB, YY, BR, YR

=> if R has B => P know he has R

=> R never say:   My hat is one of three colors

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9 hours ago, aiemdao said:

something wrong

BMAD: My hat is one of two colors

=>  P and R 's hat is : RY, RB, YY, BR, YR

=> if R has B => P know he has R

=> R never say:   My hat is one of three colors

Spoiler

If anyone has a red hat, then the other two would say their hat is one of two colors. But only BMAD made that statement. (So there are no red hats.) As you point out, that means she saw two yellow hats. So Rocdocmac and plasmid have the yellow hats. That makes BMAD's hat blue.

So when you say R would never say "three colors" you're overlooking the case that he could see a blue hat and a yellow hat.

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1 hour ago, bonanova said:

Reveal hidden contents

If anyone has a red hat, then the other two would say their hat is one of two colors. But only BMAD made that statement. (So there are no red hats.) As you point out, that means she saw two yellow hats. So Rocdocmac and plasmid have the yellow hats. That makes BMAD's hat blue.

So when you say R would never say "three colors" you're overlooking the case that he could see a blue hat and a yellow hat.

I mean because their comment are sequent . So Rocdocmac know he dont have Red ( Plasmid say one of three ) , and he cannot have B too because if he have blue , Plasmid will know he have Red :v

=>he already know he have Yellow

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5 hours ago, aiemdao said:

I mean because their comment are sequent . So Rocdocmac know he dont have Red ( Plasmid say one of three ) , and he cannot have B too because if he have blue , Plasmid will know he have Red :v

=>he already know he have Yellow

@aiemdao You're exactly right. OP should have said they made their statements simultaneously, or that they wrote their statements on slips of paper without reading the other two statements. I'm making that change in the OP now.

You get Honorable Mention recognition for your answer. Thanks, and Happy New Year!

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