# Three blue and two red hats puzzle

At a party, four people played a game. Three of them sat one behind the other, so that Abe saw Bill and Cal, and Bill only saw Cal, who sat in front and saw nobody else.

Dave had 5 hats which he showed to his three friends. Three of the hats were blue and two were red.

Now Dave placed a hat on the head of each of his three friends, putting aside the remaining two hats.

Then he asked Abe what color his hat was.

Abe said he couldn’t tell.

Bill was asked the color of his own hat, but he didn’t know for certain either.

Cal, though he couldn’t see any hat at all, gave the right answer when asked what color his hat was.

Do you know what color Cal’s hat was and how he reasoned to find out the correct answer?

Color of Cal’s hat?

Cal’s hat is blue.

If Abe sees two red hats he would know his hat is blue.

So he sees at least one blue hat. The other one can either be red or blue.

If Bill sees a red hat he knows he must have a blue hat.

But Bill doesn’t know what hat he haves so he must be seeing a blue hat.

“So he sees at least one blue hat”

There are 3 blue hats, so all 3 can be blue. This isn’t a guarantee that one of the front 2 is blue.

Abe doesn’t know which hat he is wearing, so the two in front must be either 1 red 1 blue or 2 blue. Can’t be 2 red or Abe would know his is blue.

Bill doesn’t know which hat he’s wearing, but he knows it must be 1 red 1 blue or 2 blue. If the hat in front is blue, then he would know his is red. Therefore the hat in front must be red.

Cal’s hat is red.

This is not correct.

Abe sees at least one blue hat.

If Bill sees a red hat he knows he must have a blue hat.

But Bill doesn’t know what hat he haves so he must be seeing a blue hat.

So Cal’s hat must be blue.

Yup, I inverted the logic at the end by accident – good spot.

To correct mine:

Bill doesn’t know which hat he’s wearing, but he knows it must be 1 red 1 blue or 2 blue. If the hat in front is red, then he would know his is blue. Therefore the hat in front must be blue, and Cal’s hat is red.

### Your Answer

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