Born on a Tuesday

Suppose I tell you that I have two children and one of them is a boy, what is the probability that I have two boys?

The correct answer is not 1/2 but 1/3. How can that be? Well there are four possible combinations, BB, GG, BG and GB but but at least one is a boy so you can get rid of GG. So all that's left is BB, BG and GB; and in only one of those three possibilities do I have two boys.

Now I tell you that I have two children and one of them is a boy born on a Tuesday. What is the probability that I have two boys?

Wise Men Don't Wear Hats

An emperor decides to give 100 of his philosophers a test. The philosophers will stand in line, one behind the other, so that the last person in the line sees everyone else. The emperor has 101 hats, each of a different color, and the philosophers know all the colors. The emperor puts all but one of the hats on the heads of the philosophers, one per philosopher. The philosophers can only see the colors of the hats on people in front of them. Then, in any order they want, each philosopher guesses the color of the hat on his own head. Each hears all previously made guesses, but other than that, the philosophers cannot speak. They are not allowed to repeat a color that was already announced. Each person who guesses his color wrong will get his head chopped off. If a color is guessed a second time, all the philosophers will have their heads chopped off. The ones who guess correctly go free. The rules of the test are given to them one day before the test, at which point they have a chance to agree on a strategy that will minimize the number of people who die during this test. What should that strategy be?

Bug march

A bug starts at the top-right corner of a square and marches counterclockwise around the perimeter of the square at a uniform speed. A spider starts at the bottom-right corner of the square and moves at the same speed as the bug, always in the direction of the bug. Where is the spider when the bug reaches the top-left corner? Does the spider catch the bug?