3 red hats, 3 blue hats, and 3 boxes?

Question

This problem is from the Prentice Hall Geometry book in p. 187.

Three red hats and three blue hats are packed in three boxes, with two hats to a box. There is a picture, which is that the 1st box has two red hats, the 2nd box has two blue hats and the 3rd box has one blue hat and one red hat. The boxes i just described are all labeled incorrectly. To determine what each box actually contains, you may select one hat from one box. Explain how this will allow you to determine the contents of each box. (Hint: List all possible solution; then use logic to solve.) FYI: This is for school

Answer 1

I would start with the one labeled with one blue and one red (mixed). Since the label is wrong, whatever color hat we draw out will be the correct color of *both* hats (has to be two blue or two red).

Once you know that, the other two labels are obvious.

Let's create 3 labels. I'll call them R (red), B (blue), M (mixed).
R, B, M

There are 6 possible arrangements of hats:
R, B, M --> 3 labels aligned
R, M, B --> 1 label aligned
B, R, M --> 1 label aligned
M, B, R --> 1 label aligned
M, R, B --> 0 labels aligned
B, M, R --> 0 labels aligned

So there are really only two possible ways you could mislabel the hats and have all labels be wrong:

R, B, M
-----------
M, R, B
B, M, R

If you pick from the "mixed" box and find it has blue, then you have case 1 and you know the labels are:

Red label really has both red and blue hats
Blue label has red hats
Mixed label has blue hats.

Or if you find that box has a red hat, then you have case 2:

Red label has blue hats
Blue label has both red and blue hats
Mixed label has red hats

SUMMARY OF THE LOGIC:

Pull from the mixed box. The color hat tells you the color of this box. Let's say it is blue.

The box that is mislabeled red can't be all red. And we now know it can't be all blue. So it is mixed.

That leaves the box labeled blue, which has to be red.

The same logic works in reverse if you draw a red hat from the mixed box.