Puzzle 7

(Related to the lectures of December 1st, 2007)

Once again, there are two puzzles this month, in celebration of the first-ever SMMG double-header. You can win a prize for each puzzle you solve correctly; see if you can solve both of this month's puzzles!

Problem 1 (Related to Dr. Rossmo's talk): Recently, a rare disease called "calculitis" has begun to afflict freshmen at universities and colleges across the United States. Fortunately, the disease can be treated if it is detected early. One way to determine whether a freshman has calculitis is to apply a chemical to his/her scalp; this chemical should cause the patient's hair roots to turn purple if he/she is infected. (This test is called the root test. The root test has a 98% chance of returning a correct result if the patient is not infected, and a 99.9% chance of returning a correct result if the patient is infected. Approximately 0.2% of college students become infected with calculitis during their first year of college. Answer the following questions, explaining each of your answers carefully:

(a) Suppose we choose a freshman at random and have him/her take the root test. What is the probability that he/she has calculitis, and the test determines that he/she has calculitis?
(b) Suppose we choose a freshman at random and have him/her take the root test. What is the probability that he/she does not have calculitis, but the test determines that he/she has calculitis?
(c) Suppose we choose a freshman at random and have him/her take the root test. Assuming that the test determines that he/she has calculitis, what is the probability that the student actually has calculitis?

You can find a solution to this puzzle here. Congratulations to Rina Sadun, who sent in the winning solution. We are still accepting solutions to Problem 2, which is given below:

Problem 2 (Related to Cody Patterson's talk): Answer the following questions, explaining each of your answers carefully:

(a) How many ways are there to tile a 3x5 rectangular board with 1x2 rectangular tiles? (Here, to tile a board is to cover it with tiles so that no two tiles overlap, and no tiles hang off the edge of the board.)
(b) How many ways are there to tile a 3x6 rectangular board with 1x2 rectangular tiles? (Hint: try solving this problem for smaller rectangular boards first. You may also find it useful to consider certain boards that are not quite rectangular.)

Email your answer to smmg@math.utexas.edu. The first person to send in a correct solution will win a cool prize!

Puzzle 6

(Related to the lecture of November 3rd, 2007)

There are two puzzles this time! (We had too many good ones). You will get a prize for each one of these you answer correctly (so up to two prizes).

Problem 1: Consider the standard square lattice in the plane (points (a,b) with integer coordinates) and let R denote the union of disks with centers at the lattice points and radius 1/2. What percentage of the plane is covered by R? Explain what the question (and your answer) mean as precisely as you can. What if you put instead the hexagonal lattice?

Click here to see an example of a hexagonal lattice.

You can also download a square lattice if you want to have some visual aid by clicking here.

Problem 2: This one is a two parter.

  1. Find the sum 1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + ... + 1/(2007*2008). (Hint: Try adding up the first few terms. What do you notice?)
  2. Suppose that n is a positive integer. Is it always true that 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... + 1/n^2 <= 2? If this inequality is always true, explain how you know this is the case. If the inequality is not always true, find a value of n for which it is not true.

Email your answers to smmg@math.utexas.edu. The person who sends in the first correct answer to either puzzle will get a very cool prize!

Puzzle 5

(Related to the lecture of October 13th, 2007)

Problem (also from Dr. Starbird's book, "The Heart of Mathematics") : Let S be the set of all real numbers between 0 and 1 with the property that their decimal expansions only have 0's and 7's. For example, the following numbers are elements of S:

0.7777007707070777707000....

0.00000000000070000077777770000007....

Is the cardinality of S equal to the cardinality of the set of natural numbers? Why or why not?

Solution.

Congratulations to Areen, Kaarthik, and Tapasvini. They all got a copy of the book "An Introduction to Inequalities", by Edwin Beckenbach and Richard Bellman.

Puzzle 4

(Related to the lecture of September 15th, 2007)

(This is a puzzle from Dr. Michael Starbird's book, "The Heart of Mathematics")

It's in the box: There are two boxes: one marked A and one marked B. Each box contains either $1 million or a deadly snake that will kill you instantly. You must open one box. On box A there is a sign that reads: "At least one of these boxes contains $1 million." On box B there is a sign that reads: "A deadly snake that will kill you instantly is in box A." You are told that either both signs are true or both are false. Which box do you open? Be careful, the wrong answer is fatal!

Solution.

Congratulations to Jonathan, Lamson, Areen, Garrett, Kapil, Rebekah, Khoa, and Jasmine! They all got a copy of "Math Magic Show" by Martin Gardner.

Puzzle 3

(Related to the lecture of April 7th, 2007)

Magic Triangle Problem: When you place the numbers 1-9 in the given circles so that the sum of the four numbers along each side of the triangle is the same, what values are possible for this common sum?(There can be more than one answer) Justify.

Solution.

Puzzle 2

(Related to the lecture of February 24th, 2007.)

Problem (by M. Gardner): My wife and I recently attended a party with four other married couples. At the beginning, various handshakes took place. No one shook hands with himself or herself, or with his or her spouse, or with somebody more than once. After all the handshakes were over, I asked each person, including my wife (a total of nine questions to nine people), how many hands he or she had shaken. Each gave a different answer. How many hands did my wife shake?

Solution.

Congratulations to Khalid, Ms. Gorton's Math Group and Matthew for answering this puzzle. They all got a copy of the book "Riddles of the Sphinx" by Martin Gardner.

Challenge Problem: Suppose that you have a group of people in a room. Some have met each other before, some of them have never met. How many people should there be in the room so that you can be sure that either 3 of them know each other or four of them have never met? Explain your reasoning. (Hint: you can think of this problem in terms of pictures.)

Congratulations to Khalid for solving this very hard puzzle!!!

Puzzle 1

Note: This puzzle is related to the lecture of February 3rd, 2007.

Problem: How many sets (including overlapping ones) are possible in the deck of SET?

Solution.

Congratulations to Khalid and Rina (and company), for solving this puzzler. They all got a copy of a book by Ivan Niven.