S.KRAUSE

Problems

PROBLEM 1:

The game Tetris uses seven flat pieces each made up of four squares. These are called "polynomios of order 4". Assume that you have one of each of these. Can you put them together to form a rectangle without any holes?

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PROBLEM 2:

At a dinner party aboard the Starship Enterprise three life forms are present: Humans, Klingons, and Romulans. The dinner table is a long 1xN board, and the life forms sit on one side of it, one next to another. From each life form there are more than "n" individuals present, so only a total of "n" sit at the table. The only problem is that no two humans want to sit next to each other. Let H(n) denote the number of different ways that "n" life forms can be seated at the dinner table. Assume that all humans look alike, as do all Klingons, and all Romulans.

1. What is H(1)? What is H(2)?
2. Which one(s) (if any) of the following are true, and which are false:
   1. H(n) = 3H(n-1) - H(n-2)
   2. H(n) = 2H(n-1) + 2H(n-2)
   3. H(n) = 3H(n-1) - (n-1)!
   4. H(n) = H(n-1) + 3H(n-2) + 2H(n-3)

Give adequate and complete reasoning for your answers.

- problems 1 and 2 courtesy of Prof. Shahriari at Pomona College (well, they're his problems, and I'm just copying them)

PROBLEM 3:

The volume of a solid is equal to the sum of three similar solids. The surface areas of these solids are equal to the squares of four consecutive integers. What is the surface area of the largest solid?

- problem III forwarded to me by Andrew Manies.

PROBLEM 4:

Three prisoners sit facing each other in a room. There are 5 hats: 3 black, 2 white. The prisoners are blind-folded and each prisoner has a hat placed on her head. The blind-folds are removed, and each prisoner is told her life will be spared if she can identify the color of hat on her head. The first prisoner is unable to do so. The second prisoner is unable to as well. The third says, "I know the color of the hat on my head." What color hat is the 3rd prisoner wearing?

PROBLEM 5:

Four bugs are placed at the corners of a square. Each bug walks directly toward the next bug in the clockwise direction. The bugs walk with constant speed always directly toward their clockwise neighbor. Assuming the bugs make at least one full circuit around the center of the square before meeting, how much closer to the center will a bug be at the end of its first full circuit?

- problem taken from a list of puzzles at Caltech.

PROBLEM 6:

The Cow, the Sheep, and the Harmonica: 2 men own x cows and sell them for $x per head. They then buy sheep at $12 per head. The income ($x²) wasn't divisible by 12, so they bought a lamb with the remainder. Later they divide the flock so that each has the same number of animals, but the man with the lamb was short-changed. So to even things up, the other man gave him his harmonica. How much was the harmonica worth?

- problem 6 courtesy of Prof. Shahriari at Pomona College.