Inductive vs Deductive Reasoning - Homework 1 | MATH 300 | Assignments Algebra

UMASS AMHERST MATH 300 SPRING ’09, F. HAJIR

HOMEWORK 1: PROBLEM SOLVING, INDUCTIVE VS. DEDUCTIVE REASONING, AN

INTRODUCTION TO PROOFS

1. Suppose you are given eight identical-looking and identical-feeling balls. You are also

given a balance scale. (By placing balls on the two pans of the balance, you can establish

whether the two collections weigh the same, and if not, which collection of balls is heavier

than the other). You are given the information that of the 8 balls, 7 of them weigh exactly

the same and one of them is slightly heavier than the others. You are allowed to use the

balance twice, no more.

Proposition. There is a strategy guaranteed to reveal the identity of the heavy ball in just

two weighings.

Determine the truth or falsity of this Proposition. If you believe it is true, describe a

strategy and explain exactly why it is guaranteed to work. If you do not believe it to be

true, then give a convincing argument for the lack of a strategy.

2. Olga, Laurel, Serge, and Zoltan have been hired as coaches for basketball, soccer,

volleyball, and swimming. Zoltan, whose sister was hired to coach basketball, has never

heard of Mia Hamm. Laurel doesn’t like sports that involve balls. Do you have enough

information to determine which sport each person was hired to coach? If yes, give the coach-

sport correspondence. Be sure to give a detailed and careful explanation of the reasoning

process by which you arrived at your conclusions.

Note. In case you are not sure how Mia Hamm fits into the picture: I should tell you

that in this problem, you can assume that anyone who has never heard of Mia Hamm is not

the soccer coach. Also, for those who do not know, Olga and Laural are women; Serge and

Zoltan are not.

3. (a) Consider the pattern formed by the sums of the first nintegers.

1=1

1 + 2 = 3

1 + 2 + 3 = 6

1 + 2 + 3 + 4 = 10

Try to explain with a picture why the numbers 1,3,6,10,15, ... are called triangular num-

bers. [My own pet name for them is “bowling” numbers – can you explain that too?]

(b) The first bowling number is 1. Can you predict the tenth bowling number? Make a

table of the first twelve bowling numbers. Can you determine hundredth bowling number?

[When Gauss was six, he did this in a few seconds, astounding (and perhaps annoying?) his

teacher.] Can you do it without writing them all down up to the hundreth? By messing

around with a table of some bowling numbers, see if you can come up with a formula for the

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UMASS AMHERST MATH 300 SPRING ’09, F. HAJIR

HOMEWORK 1: PROBLEM SOLVING, INDUCTIVE VS. DEDUCTIVE REASONING, AN INTRODUCTION TO PROOFS

Suppose you are given eight identical-looking and identical-feeling balls. You are also given a balance scale. (By placing balls on the two pans of the balance, you can establish whether the two collections weigh the same, and if not, which collection of balls is heavier than the other). You are given the information that of the 8 balls, 7 of them weigh exactly the same and one of them is slightly heavier than the others. You are allowed to use the balance twice, no more. Proposition. There is a strategy guaranteed to reveal the identity of the heavy ball in just two weighings. Determine the truth or falsity of this Proposition. If you believe it is true, describe a strategy and explain exactly why it is guaranteed to work. If you do not believe it to be true, then give a convincing argument for the lack of a strategy.
Olga, Laurel, Serge, and Zoltan have been hired as coaches for basketball, soccer, volleyball, and swimming. Zoltan, whose sister was hired to coach basketball, has never heard of Mia Hamm. Laurel doesn’t like sports that involve balls. Do you have enough information to determine which sport each person was hired to coach? If yes, give the coach- sport correspondence. Be sure to give a detailed and careful explanation of the reasoning process by which you arrived at your conclusions. Note. In case you are not sure how Mia Hamm fits into the picture: I should tell you that in this problem, you can assume that anyone who has never heard of Mia Hamm is not the soccer coach. Also, for those who do not know, Olga and Laural are women; Serge and Zoltan are not.
(a) Consider the pattern formed by the sums of the first n integers.

Try to explain with a picture why the numbers 1, 3 , 6 , 10 , 15 , ... are called triangular num- bers. [My own pet name for them is “bowling” numbers – can you explain that too?] (b) The first bowling number is 1. Can you predict the tenth bowling number? Make a table of the first twelve bowling numbers. Can you determine hundredth bowling number? [When Gauss was six, he did this in a few seconds, astounding (and perhaps annoying?) his teacher.] Can you do it without writing them all down up to the hundreth? By messing around with a table of some bowling numbers, see if you can come up with a formula for the 1

2 HOMEWORK 1: PROBLEM SOLVING, INDUCTIVE VS. DEDUCTIVE REASONING, AN INTRODUCTION TO PROOFS

nth bowling number. I’m not asking you to prove the formula is correct, but if you want to do that, give it a whirl! (c) Twenty people come to a fancy dinner. Each guest shakes hands with every other guest exactly once. How many handshakes occur? (d) Twenty points are marked on the perimeter of a circle. The line segment joining every pair of distinct marked points is drawn. How many line segments are drawn? (Try it as a thought experiment or as an actual activity). (e) Now the number of people coming to the fancy dinner keeps fluctuating, so we just want to call the number of guests x and wish to have a formula H(x), where H(x) is the number of handshakes among x guests. What about where H(x) is the number of line segments among x marked points on the circle?

Elizabeth Rossolimo has 44 pennies and ten pockets in her vest. She wants to put all her pennies in her pockets in such a way that she has a different number of pennies in each pocket. Can she do it? If yes, tell Elizabeth how, if not, explain to her (gently but cogently) why she cannot.
Consider a clock with line-thin hour-hand and minute-hand that move perfectly con- tinuously. At noon and then again at midnight, the two hands line up perfectly. (a) How many times between noon and midnight (counting midnight, but not noon) do the two hands line up again? (b) Determine the exact times at which the two hands line up. Explain your reasoning very carefully.

Extra Credit Problems.

AN IMPORTANT NOTE ABOUT EXTRA CREDIT PROBLEMS. These problems are for your amusement and edification and to challenge or push you. Don’t look here for an easy way to ameliorate your grade: a better way of doing that would be to concentrate more on other aspects of the course. There is no expectation here about whether you “should” be able to solve even a small part of any of these problems. In some cases, the solutions here are difficult and beyond this course entirely. In some cases, I do not myself know the solution to the problems stated here, and in fact the solution may be as yet unknown to homo sapiens. And in other cases, the solution may come to you easily, in a flash. The point here is to present what I think are interesting problems and see how far you can run with them, for your own entertainment and growth. You will receive a certain number of “bonus points” depending on the difficulty of the problem and how far you travelled into the solution. At the end of the semester, the student with the largest number of bonus points will receive a fabulous (sur)prize. I will be reluctant to reveal too many solutions too soon, as I want people time to hammer away at their favorites for quite some time.

A. Consider a generalization of Problem 1. Namely you are given n balls, 1 of which weighs slightly more than all the others, while the other n − 1 are identical. Let s(n) be the smallest number k ≥ 0 with the following property: if you are allowed to use the balance scale k times, then there is a strategy that will always allow you to identify the heavy ball.

Inductive vs Deductive Reasoning - Homework 1 | MATH 300, Assignments of Algebra

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UMASS AMHERST MATH 300 SPRING ’09, F. HAJIR