Can anyone help me with some logic problems?

Question

These problems are for my math homework, and I really don't know how to get the answers. Please help?

1: Three red hats and three blue hats are packed in three boxes with two hats to a box. The boxes are all labeled incorrectly. To determine what each box actually contains, you may select one hat from one box, without looking at the contents of the box. Explain how this will allow you to determine the contents of each box.

2: Alan, Ben, and Cal are sitting in order. Alan is behind Ben who is behind Cal. They all have their eyes closed. Three hats are placed on their heads from a box they know contains 3 red and 2 blue hats. They open their eyes and look forward. Alan cannot deduce what color hat he is wearing, and neither can Ben. Cal knows what color hat he is wearing. How does Cal know what color his hat is?

Answer 1

1)  You left it out, but I presume the labels are "Two Red", "Two Blue" and "One of Each".  Go to the "One of Each" box and take out a hat.  If it's red, then you know that the other hat in the box must also be red (because the box is mislabeled".  Therefore you have the "Two Red" box.

You know both of the other boxes must be mislabeled as well.  If the box that had the "Two Red" sign on it were the real "One of Each" box, then the "Two Blue" sign would have been correct.  Therefore, the box that had the "Two Red" sign on it must actually be the "Two Blue" box, and the one with the "Two Blue" sign must be the "One of Each" box.

It works the same way if you find a blue hat in the box.  Just change "Red" to "Blue" above.

2)  We'll assume they're all perfect reasoners.

If Alan had seen two blue hats, he'd have known his hat must be red.  Since he didn't know, he didn't see two blue hats.  He must therefore have seen two red or a red and a blue.

Given that, if Ben had seen a blue hat, knowing what happened to Alan, he would have known that his own hat must be red.  He doesn't know, so he must not have seen a blue hat.  Therefore, he must have seen a red hat.  Cal, then, knows that he's wearing a red hat.