FYS: Puzzles and Paradoxes--Problem Set 9

Complete exactly three of the following five problems.

1. A bookworm eats from, and including, the first page of a bound paper encyclopedia to, and including, the last page.  The bookworm eats in a straight line.  The encyclopedia consists of ten 1000-page volumes and is sitting on a bookshelf in the usual order. 

Not counting covers, how many sheets of paper does the bookworm eat through? Justify your answer.

2. Let A be the count of how many 0's are in a given number. Let B be the number of 1's in a given number, and so on.

Is it possible for ABCDEFGHIA to itself express the given number? Justify your answer.

3. King Egbert likes to order his knights around. One day, he has his 100 knights form a 10 by 10 square with 10 knights in each row and 10 knights in each column. He then asks the knight who is the tallest of the shortest people in each column to raise his hand. The knight, Sir Fredric, does so. He then asks the knight who is the shortest of the tallest people in each row to raise his hand. A different knight, Sir Gallis, raises his hand.

Is it possible to determine who is taller between Sir Gallis and Sir Frederic? Justify your answer.

4. The Moderator takes a set of eight stamps, four black and four white, known to the Logicians, A, B, and C, and loosely affixes two to the forehead of each Logician so that each Logician can see all the other stamps except those two in the Moderator's pocket and the two on her own head.  He asks them in turn if they know the colors of their own stamps:
  A: "No", B: "No", C: "No", A: "No, B: "Yes"

Is it possible to determine the distribution of all the stamps? Justify your answer.

5.[Bonus Point Problem] Two numbers, not necessarily different, are chosen from the range of positive integers from one to nine inclusive. The product of the numbers is given to Mr. P, and the sum of the numbers is given to Mr. S. The following conversation ensues:

          P: I do not know what the numbers are.
          S: I do not know what the numbers are.
          P: I do not know what the numbers are.
          S: I do not know what the numbers are.
          P: I do not know what the numbers are.
          S: I do not know what the numbers are.
          P: I do not know what the numbers are.
          S: I do not know what the numbers are.
          P: I know what the numbers are.

          Is there sufficient information for you to determine what the two numbers are? Justify your answer.

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