M302, Spring 2005 (Smith)

INFORMATION FOR FIRST EXAM

Date: Friday, February 18
Time: Usual class time (12 noon - 12:50 p.m.)
Place: WEL 1.308 (Welch Hall is at the northeast corner of 24th St. and Speedway; click to see map and picture. Room 1.308 is down one floor from the street level.)

What you may and may not bring to the quiz:

You should bring a pencil, eraser and scratch paper.
You may bring a non-programmable calculator.
You may not bring notes, your textbook, a programmable calculator, or any other programmable device such as a Palm Pilot.

Quiz procedures: Please sit every seat in the auditorium.

Quiz coverage: The quiz will cover Chapter 1 and Sections 1, 2, and 4 of Chapter 2.

Quiz format: The quiz will be "semi-take-home." By that I mean that the quiz questions will be very similar to some selection from the following list of questions. By "very similar" I mean, for example, that the numbers might be changed or the wording might be different. (The retake at the final exam time will be a slightly different selection, but also taken from the questions below.) You are allowed to get help with these questions from anyone except me or the TA before the exam. (You may ask me or the TA questions about anything assigned for class or covered in class, but not about the questions below or about very similar questions, unless they have been assigned in the reading, assigned as homework, or discussed in class.) As stated above, you may not bring any notes to the quiz. Also, be prepared for the questions on the quiz to be in a different order than the corresponding questions below. There might be an optional bonus problem not like any on this list.

Possible quiz questions:

Reminder: Unless told otherwise, you need to explain how you got your answers to receive full credit.

1. Write down the first ten Fibonacci numbers.

2. Express the number 67 as a sum of distinct, non-consecutive Fibonacci numbers.

3. The twelfth and thirteenth Fibonacci numbers are 144 and 233. What is the fourteenth Fibonacci number? What is the fifteenth Fibonacci number? What is the eleventh Fibonacci number? Explain how you can obtain your answers from the given information without using any Fibonacci numbers before the twelfth.

4. If the number of spirals going in one direction on a sunflower seed head is 12 13, what would you expect the number of spirals going in the other direction on that same sunflower seed head to be? Why? (Your answer should be of the form, "Either ____ or ____.)

5. Explain how you know that there are at least two students on campus who weigh exactly the same, to the nearest ounce. (There are 16 ounces in a pound.)

6. Explain the winning strategy for the game Dodge Ball (as described in Chapter 1 of the textbook.) Would the strategy work for a different-sized board? For an infinitely big board?

7. There are 24 socks in a drawer. Ten of them are red, eight are blue, and six are white. If the socks are all mixed up and you pull 9 of them out, one at a time, without looking, will you be guaranteed to have four socks of the same color? Explain. What if you pulled out ten socks?

8. You are given eight coins which all look identical. However, seven coins are genuine and one is counterfeit. You know that the real coins all weigh exactly the same and the fake coin is slightly heavier. You are allowed to use a balance scale exactly twice. Explain how you would identify the counterfeit coin.

9. The UPC (bar code) for Progresso Black Beans is
       0    41196   0212 6
Unfortunately, the second to last digit got smudged. What should it be? Explain.

10. Today is Wednesday. What day of the week will it be in 129 days? What day of the week was it 219 days ago? Explain.

11. Find the smallest positive number that is equivalent to the given number or expression mod the given number:

   a. 3 x 4 + 2 x 6 (mod 10)

   b. 8 (mod 6)

    c. 243 x 4 + 5642 x 6 (mod 10) [Hint: There's an easy way to do this.]

    d. -3 (mod 5)

12. What is the remainder when 14²⁵¹ is divided by 13? Explain. [Hint: You won't be able to calculate 14²⁵¹ on your calculator, so think mod 13. It should help to look at 14² first.)

13. Recall that F_n stands for the n^th Fibonacci number.
    a. Write out the first eight numbers of the form F_n + F_n+1. (So the first number in your list would be F₁ + F₂ = 1 + 1 = 2.)
   b. By examining the list of numbers in part a, give another method (that is, besides calculating from the formula F₉ + F₁₀) for finding the next number in the sequence.

14. In a standard deck of 52 cards, what is the smallest number of cards you must draw to guarantee that you will have 5 cards of one suit? (A standard deck of cards has four suits -- hearts, diamonds, clubs, and spades -- with 13 cards in each suit.)

15. Eighty thousand people attended the UT versus A&M football game. The fans of both teams were so happy, they decided to organize a party each day for a year. They decided that on each day, anyone with their birthday on that day would return to the stadium at noon to celebrate. Explain why at least one party would have more than 200 people.

16. Each box of animal crackers contains two servings; each serving consists of exactly 12 crackers. There are exactly 18 different shapes of crackers. Are there always two crackers of the same shape in each box/ Explain why or why not.

17. You have two measuring cups. One holds exactly 4 ounces of water and the other holds exactly 3 ounces of water. There are no markings on the cups and you are not able to mark the cups at all. You are given a huge bucket of water. Is it possible to measure and place exactly 2 ounces of water in to the big cup? If so, carefully explain why. If not, carefully explain why not.

18. July 9, 1949 was a Saturday. What day of the week was July 9, 1950?

19. You started a long mathematics exam at 2:00 p.m. You were told that you could work as long as you liked. You worked 487 hours straight. At what time of day did you finish?

20. You have a carving that is 3 ft. long and 6 inches wide. You want to pack it in a box to ship to a friend. You go the store to buy a box to put the carving in, but you don't take the carving with you, just the measurements. The store has only two sizes of boxes, one that is 1 ft. wide and 2.5 ft. long, and the other that is 2 ft. wide and 2 ft. long. Each box is a little over 6 inches high, so you think you might be able to fit the carving on edge in one of the boxes. Can it fit in either box? If so, which one? Or both? How do you know?