[Math Lair] Miscellaneous Puzzles

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  1. At a certain convention, there were 100 politicians. We know the following two statements are true:
    1. At least one politician was honest.
    2. Given any two politicians, at least one of the two was crooked.
    Can it be determined from the above facts how many of the politicians were honest and how many were crooked?
  2. Three subjects, A, B, and C, are all perfect logicians, and are all aware that both of the other two are also perfect logicians. The three were shown seven stamps: Two red, two yellow, and three green. They were then blindfolded, and a stamp pasted on each of their foreheads. When the blindfolds were removed, A was asked, "Do you know one colour that you definitely do not have?" A answered, "No." B was then asked the same question and also answered, "No." Is it possible, from this information, to deduce the colour of any of the three stamps placed on any of A, B, or C?
  3. A bottle of wine cost $10. The wine was worth $9 more than the bottle. How much was the bottle worth?
  4. Suppose that you and I have the same amount of money. How much money would I have to give you such that you have $10 more than I do?

Answers are on the answer page.