# HAVING FUN WITH PROBLEM-SOLVING

Having Fun With Problem-Solving

[A Memoir
By Roy Wysnewski]

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted by any means without written permission by the author.

CONTENT

INTRODUCTION

PROBLEMSCONVENTIONAL

1. Math/Science

2. Numbers

3. Numbers (Age)

4. Math/Science

5. Math (Magic)

6. Numbers (Percentages)

7. Logic

8. Science

9. Logic

10. Cards

11. Algebraic Math

12. Numbers/Math

13. Logic

14. Math

15. Numbers/Logic

16. Numbers/Math

PROBLEMS – ‘FAR OUT

INFINITY

17. Infinity/Numbers

18. Infinity/Numbers

19. Infinity (Pi)

20. Infinity/Math/Physics

UNIVERSE

21. Universe/Astronomy – Cosmology

1. Math/Numbers

2. Numbers

3. Numbers (Age)

4. Math/Science

5. Math (Magic)

6. Numbers (Percentages)

7. Logic

8. Science

9. Logic

10. Cards

11. Algebraic Math

12. Numbers/Math

13. Logic

14. Math

15. Numbers/Logic

16. Numbers/Math

17. Infinity/Numbers

18. Infinity/Numbers

19. Infinity (Pi)

20. Infinity/Math/Physics

21. Universe/Astronomy – Cosmology

HAVING FUN WITH PROBLEM-SOLVING

INTRODUCTION

Problem-solving, hereafter identified as P-S, is reported to be among the six most important life skills. And while we frequently do P-S, we generally don’t become conscious of the process until our early school years. The reason for this is that the skills involved like being analytical – the ability to separate a problem into meaningful parts, and methodical – the ability to develop a procedure for achieving an answer to the problem – take time to develop and recognize. There’s no set time for this to happen as some individuals gain P-S awareness sooner than others.

For me, I didn’t become aware of my P-S skills until about the sixth grade when I was introduced to increasingly more difficult mathematics and science problems. Then, at college, the successful completion of different math and science courses having many complex problems sharpened those skills. I became very good at P-S during my forty-year career in Nondestructive Testing in which I was frequently confronted with a variety of difficult managerial and technical problems.

We encounter all kinds of P-S during our lifetime. Some are rather easy/mundane ones like: How should I dress on this cool and rainy morning? Others, as I learned in college and during my professional career, can be difficult. Typical examples might include: How to derive the math constant ‘Pi’; Determining when and how mankind will colonize the planet Mars; and, Define the size of the universe – is it finite or infinite?

Many problems we encounter are of a 'serious' nature and there is little enjoyment in solving them. However, occasionally we are exposed to problems on the lighter side - problems whose solutions can be associated with having a 'fun' experience. Earlier this year while browsing Facebook posts, several math type problems caught my attention. Two such problems are shown below. The example on the left happens to be a trick numbers/math problem and according to the author of the Facebook post, only 10% of those who responded got the correct answer. Those few either guessed the correct answer or used P-S skills. Now, I find this kind of P-S to be fun and I suspect, judging from the response to these Facebook posts (and others like it), there are many of you out there in the cyber universe who also find enjoyment in this type of P-S.

While working with these entertaining problems found on Facebook, it occurred to me that I have in my files a number of similar ones that I accumulated over the years. In addition to numbers/math problems, my collection includes unique and sometimes ‘tricky’ logic, physics and even playing card problems all of which I thoroughly enjoyed solving I So, the thought occurred to me – why not share the fun solving these neat problems with others via an e-book memoir about P-S.

The first sixteen problems are of a conventional nature – problems that may be more typical of those you’ve previously seen while the remaining ones are dedicated to what I like to call ‘far out’ – those that can be philosophical in nature and do not necessarily have exact solutions. But more about this later. Problems will be numbered, then labeled as to their category, i.e., numbers, math, science, logic, etc. Each problem is described by a Problem Statement {what are the details of the problem} followed by a Question {what needs to be solved/answered, and in some cases, a description of the solution process}. Solutions/answers for all the problems can be found starting on page 16.

Have Fun!

PROBLEMSCONVENTIONAL

1. Math/Science

Problem Statement: This problem, first posed by my sixth grade teacher, was one of several problems that helped foster my interest in math and science.

The problem states that you are driving an automobile from Point A to Point C, a distance of two miles. You set a goal to average 60 miles per hour from A to C. After starting out, you reach a mid-point B (one mile) and ascertain that you averaged 30 miles per hour for that first mile.

Question: What average speed must you drive from B to C to assure an overall average speed of 60 miles per hour from A to C?

2. Numbers

Problem Statement: I shared this neat numbers problem with my children and grandchildren several years ago. Now it’s time to share it with others.

The problem states that you give someone you know a number, as for example 27. Next, ask them to multiply that number by any number they wish, again as an example, they multiply by the number 5 (this must be a whole, non-zero number). The product they get is 135. Give them another number, say 18, to add to what they already have. Their final number will be 153. Finally, ask them to circle one digit from their resulting number, then read back to you all the non-circled digits (non-zero). Let’s suppose they circle the number 3 and give you the other two digits – 1 and 5. You can then surprisingly identify the circled digit.

Question: How do you know what the circled digit is?

3. Numbers (Age)

Problem Statement: This problem is unique because you determine another person’s age simply by them manipulating a few numbers then at the end giving you a three-digit answer.

Note: they may need a calculator to do the number ‘crunching’.

The problem states that you tell the other person to pick the number identifying how many ‘ best friends’ they have – this number must be greater than 1 but less than 10. Ask them to multiply that number by 2. Next, have them add the number 5, then multiply by the number 50. And, if the year you do this problem is 2016, and the person has not yet had their birthday, have them add to their total the number 1765 – if they have had their birthday, add 1766. Should you be working with this problem in subsequent years, add 1 to each of these numbers per year. Finally, ask them to subtract from their total number the four-digit year they were born and give you the resultant number. This will be a three-digit number from which you tell them their age.

Question: How do you know which of the three digits represents the person’s age?

4. Math/Science

Problem Statement: Another challenging problem from my sixth-grade class was to measure a certain distance {from A to B) in the class-room, then determine if that represented the shortest distance from A to B. Presenting this problem in a different context, consider that I take a special non-stop, direct airline flight from Sarasota, Fl. to San Francisco, Ca., a distance determined from the 2015 Rand McNally Road Atlas to be 2,832 miles. After receiving the airlines free ‘Sky Miles’ for this trip, I discovered that I earned fewer miles then I expected because the airlines claim that the straight-line (shortest) distance between the two cities is actually 2,773 miles.

Questions: What is this shorter distance? Why the discrepancy? Was the airlines fair?

5. Math (Magic)

Problem Statement: This is another magical type numbers/math problem in which manipulation of two separate parts of your telephone number results in your complete number. You will need a calculator. The problem states that you follow these eight mathematical steps:

1. Key in the first three digits of your phone number (NOT the area code)

2. Multiply by 80

4. Multiply by 250

7. Subtract 250

8. Divide resulting number by 2

Question: Do you recognize the answer? Will this ‘magical, problem work with any telephone number? Demonstrate.

6. Numbers (Percentages)

Problem Statement: Numbers can be substituted for letters to form different words. For th is problem, the letters in the words add up to important percentage numbers. The problem states that specific questions about percentages can be answered using the following mathematical formula: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z is represented as: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 26 (where the numbers are percentages). Now, we've been told that to be successful in life, we need to give 100% effort. Some even say we should give more than 100%.

Question: Using the formula provided above, determine what words best describe the kind of effort needed to produce increasing percentages of 83, 94, 100 and finally 103?

7. Logic

Problem Statement: Here are four ‘cute’ logic problems.

a. Continue the following sequence in a logical way: MT WT

b. Correct this formula with a single pencil or pen stroke: 5 + 5 + 5 = 550

c. Write anything here:

d. Draw a rectangle (not a triangle) with three straight lines.

Question: What are the answers to these four problems?

8. Science

Problem Statement: This may be a difficult science problem for some because it requires specialized knowledge about the laws of physics. The problem states that you are in a row boat floating in a pond. You have with you in the boat two items --- a boulder and a very accurate measuring stick. First, you measure the ponds depth, then you roll the boulder out of the boat and into the water. Next, you take a second depth measurement.

Question: After the boulder is submerged in the pond, does the water level rise, fall or remain the same? Explain the rationale for your answer.

9. Logic

Problem Statement: I found this to be one of the best logic problems ever! The problem states that you are hiking down a road towards your destination --- a town. You come to a fork in the road and need to make a decision. Adjacent to the fork is a house occupied by two brothers. It is a known fact that one brother always lies and the second brother always tell the truth. You knock on their door and both brothers respond.

You ask one brother – not knowing whether that brother is, or isn’t the liar – one question and from the answer you immediately know what path to take to town.

Question: When confronting the two brothers, what ONE question do you ask either brother that guarantees the correct answer you need?

10. Cards

Problem Statement: This is an interesting ‘slight of mind’, not the typical ‘slight of hand’ card trick. The problem states that you look at the six playing cards shown below, then pick out one card and concentrate on that card for a couple of seconds.

Next, turn the page and you will find that I removed the card

you selected.

Question: How is this possible?

11. Algebraic Math

Problem Statement: This is an interesting, but somewhat difficult, algebra problem. The problem states that there are two equations involving the two variables --- X & Y:

4X-Y = 3V + 7

X + 8V = 4

Question: What is the numerical product of X times Y?

12. Numbers/Math

Problem Statement: This is the first of two numbers/math problems cited in the Introduction.

1 + 1 + 1 + 1

1 + 1 + 1 + 1

l + lX O + l = ?

Question: What is the net result (=?)

13. Logic

Problem Statement: I’ve only done a few logic problems of this type, but this one was definitely the most difficult to solve.

Hope you find it easier! The problem states that during a solo sail boat tour of the Caribbean Islands, I stopped at an island to replenish my supply of food and drinking water. Upon meeting some of the island’s residents, I discovered that obtaining supplies wasn’t going to be easy because I didn’t know who to trust. It seems everyone on the island belonged to one of three tribes – the Knights who always tell the truth, the Knaves who always lie, and the Knarks who make true and false statements, but never make two of the same kind in a row. The five islanders responsible for providing my supplies were introduced to me. My problem was to determine, based on two statements that each made, which tribe each islander belongs and then identify the one truthful individual. Here are the names of the five islanders and their statements.

Sabrina said :

1. Wilma is a Knark.

2. At least three of us are Knaves.

Teddy said:

3. Neither of Sabrina’s statements are true.

4. Virgil is a Knark.

Ursula said:

5. At least one of Teddy’s statements is true.

6. At least two of us are Knarks.

Virgil said:

7. Both of Ursula’s statements are true.

8. At least three of us are Knarks.

Wilma said:

9. Neither of Virgil’s statements is true.

10. Teddy is a Knark.

Question: Which islander is the truthful one (a Knight)? What tribes do the other islanders belong?

14. Math

Problem Statement: This is an historically popular algebraic word problem. The problem states that a man is twice as old as his daughter and the daughter is twice as old as her brother. In twenty-two years the son will be half his father’s age.

Question: How old is the daughter?

15. Numbers/logic

Problem Statement: This is a challenging numbers series {logic) problem.

2,4,5,10,12,24,27,.1. ?

Question: What are the last two numbers. in this sequence?

16. Numbers/Math

Problem Statement: This is the second number/math problem cited in the Introduction.

1+4=5

2 + 5 = 12

3 + 6 = 21

8 + 11 =?

Question: What is the missing{?) number? Show how you arrive at this answer.

FAR-OUTPROBLEMS

Since my early college days and my introduction to calculus, I’ve had an affinity for ‘far out’ type problems related to abstract subjects like infinity and the universe. Consequently, I decided to dedicate this section of my memoir to problems specifically involving those two subjects. Unlike the first sixteen problems which had exact answers, answers to the remaining problems may be open to some interpretation – your interpretation.

INFINITY

Infinity is an important subject in mathematics but it is not a real number. Instead, it is an idea about something that has no known end. It is represented by the symbol 00. In our ‘down to-earth’ corner of the universe in which everything seems to

‘ .

have a beginning and an end, we seldom think about something that may be endless! But, guess what? There are many examples of endless or infinite circumstances. The four problems that follow are such examples and they were developed from the elements of a well-known mathematical equation – e (i)(pi) + 1 = 0 – an equation many mathematicians affectionately refer to as the “God’s equation 1

I found these problems fun to work with and I hope you do too.

17. Infinity/Numbers

Problem Statement: This problem, which involves the two natural numbers O & 1 from the “God equation 11 and the number sequence that follows them – 2,3,4,5, ……………, is one that elementary school students are presented when they first learn to count. Natural numbers are so basic to mathematics that, in the words of German mathematician Leopold Kronecker (1823-1891), “God created the natural numbers, everything else is man’s handiwork”.

Question: A question that elementary school children are not asked to answer is the one I now ask you. Do you believe there is some ‘far out’ number that terminates the sequence or does it continue to become endlessly (infinitely) larger? What is the largest number you are familiar with?

18. Infinity/Numbers

Problem Statement: Another example of infinity experienced in everyday life evolves from a common mathematics problem – dividing 1 by 3. The resulting number 1/3 is finite. But written as a decimal number the digit 3 repeats endlessly : 0.3333…

Question: What do you believe to be a reasonable, yet acceptable answer to the question: What is 1/3 expressed as a decimal number? How did you arrive at this answer?

19. Infinity (Pi)

Problem Statement: The picture on the front cover (top) of this memoir portrays the calculation of the area of a circle. This is a common geometry problem, one that elementary students do all the time. I’m using this example to explain why Pi, the mathematical constant used to calculate a circle’s area, is such an interesting subject. In addition to being an element in the “God’s equation”, Pi represents the ratio of the circumference of any circle to the diameter of that circle. Pi also is an irrational number and as such has values of 3.14…., or 3.14159……, or perhaps 3.141592653589793238….. In fact, to show how crazy this constant’s value is, we now know, using state of the art computers, more than the first six billion digits of Pi I These facts illustrate that the numbers in this irrational number series representing Pi are seemingly endless and are sometimes said to be infinite. Returning to the problem at hand, consider a circle whose diameter is SO inches.

Question: Using the equation – Area= Pix {diameter/2) squared – what is the area in square inches? What value of Pi did you use? Is your answer exact, or is it open to interpretation? Why? And explain.

20. Infinity/Math/Physics

Question: How do we know that x -, and gamma rays cannot be stopped completely and that they continue on endlessly? What precautions are taken to keep us safe from man-made x & gamma radiation?

UNIVERSE

During the introduction, I mentioned a problem having to do with the universe – its size and age. Here is that problem.

21. Universe/Astronomy – Cosmology

Problem Statement: The number one problem confronting astronomers, cosmologists and astrophysicists during the past century relates to the size and origin (age) of the universe – is it finite (estimated to be 13.7 billion years) or infinite and did it begin with a ‘big bang’ or has it been around forever? Until the 1980s, most scientists supported a ‘big bang’ theory. That is, the universe basically exploded from nothing about 13.7 billion years ago and has expanded ever since . Then, a prominent astronomer/cosmologist from Cornell University by the name of Carl Sagan injected some new and very contemporary thinking into the debate. In 1980, when he introduced his TV series Cosmos, he is quoted as saying: “The universe is all that is or

• ever will be^11^ More recently, a cosmologist by the name of Lawrence Krauss, who supports the ‘big bang’ theory, has an interesting alternate theory about the universe’s beginnings. In a book entitled ‘A Universe from Nothing’ he writes: “The confirmation of the Higgs boson refines our understanding of the relationship between seemingly empty space and our existence”. And, framing this statement in a slightly different way, he later writes: “Hidden in what seems like empty space – indeed, like nothing – appears to be the very elements (dark energy) that allows for our existence”. Also, during the past decade a number of well-known theorists and cosmologists including Stephen Hawking introduced a controversial and bizarre ‘multiverse’ hypotheses. They proposed that our universe (cosmos) in effect becomes a multiverse , a huge eternally growing irregular shape consisting of many different universes with different properties.
p<>{color:#000;}. Questions: What do you believe the size and age of the universe to be? Is the universe expanding into an area of dark energy, another universe, or perhaps nothing?

1. Math/Science

Answer: Many people look at this problem and quickly deduce that one must go 90 miles per hour (mph.) from B to C because 90 mph from B to C plus 30 mph from A to B equals 120 mph divided by 2 equals 60 mph from A to C.

WRONG DEDUCTION!

To correctly solve this problem, we need to use the time-distance-speed relationship we learned in elementary school math/science class. That is:

(1) Time (minutes)= Distance (A – B miles)/Speed (mph)

- 1 mile/30 mph

= 1 mile-hour/30 miles

= 1/30 hour= 2 minutes.

(2) Time (minutes)= Distance {A to C miles)/Speed

= 2 miles/ 60 mph

= 1 miles/ 30 mph= 2 minutes.

The answer is that it requires 2 minutes to go from A to C, yet it took 2 minutes to go from A to B – therefore it is IMPOSSIBLE!

2. Numbers

Answer: The key to this problem is the number ‘9’. Note in the example used, the digits within each number add up to 9 or, are a multiple of 9. We began with 27 {2+7 = 9), multiplied by 5 resulting in 135 (1+3+5 = 9). By adding 18 (1+8 = 9}, we obtained a final number of 153 (1+5+3 = 9). Finally, the digit 3 was circled and the remaining numbers 1 & 5 were given to you. So, all you did was add 1 + 5 = 6, and subtract 6 from 9 and lo and behold – the circled number is 3.

Always remember, the number you give the other person must have digits totaling 9 or, a multiple of 9. You can then instruct them to multiply by any non-zero number. If you extend the problem by asking them to add or subtract a number, that number’s digits also must total nine or be a multiple of nine.

The math rule that makes this number problem possible is called “digital roots”.

3. Numbers (Age)

Answer: Let’s say I am the person you ask to solve this problem. I write down 5 as the number of my very best friends. I then multiply 5 by 2 giving me 10. Next, I add 5 giving 15, then multiply by 50 resulting in 750. Because I’ve already had my birthday this year, I add 1766 to 750 giving 2516. Finally, I subtract 1935 (the year I was born) and get a three-digit answer equal to 581. The first digit, 5, of my answer is my number of best friends and of course, 81 is my age.

This magic number problem works with any number of best friends between 1 and 10 but only when using the -X2, + 5, X SO math operatives. Here's how it works:

4. Math/Science

Answer: The chord (A-C) is determined as follows:

A-B-C is the ‘around the earth’s curvature’ distance from Sarasota, Florida to San Francisco= 2,832 miles (see figure on page 19). The earth’s circumference is 24,863 miles.

Therefore: A-B-C /circumference= Angle ADC/360 deg.

Angle ADC = 2,832 × 360 deg./24,863 = 41 deg. And, the chord A-C = 2 radius (DC) sine (A-D-C)/2 = 2(3,959) sine (A-D-C)/2 =

2(3,959) sine 41 deg./2 = 7,918 × 0.35 = 2,773 miles. The airlines is correct in saying the shortest distance from Sarasota to San Francisco is chord A-C. But is the airlines being fair?

Hardly! When we travel this route, we take the A-8-C (partial earth’s circumference) route and as we’ve just learned, it is a longer distance (and the only route we can take) and we should have been rewarded with 59 additional points.

5. Math (Magic)

Answer: I recognize the answer as my telephone number. Yes, this magic math works with any telephone number as demonstrated below:

And, there you have it! The final answer is the original seven digit number.

6. Numbers (Percentages)

H+A+R+D+W+O+R+K = 8+1+18+4+23+18+11= 83%

K+N+O+W+L+E+D+G+E = 11+14+18+23+12+4+7+5 = 94%

A+T+T+l+T+U+D+E = 1+20+20+9+20+21+4+5 = 100%

B+E+l+N+G H+A+P+P+Y = 2+5+9+14+7+8+1+16+16+25 = 103%

The conclusion is that by applying 'hard work', our 'knowledge', and a good 'attitude' to our life's endeavors we achieve 100% effort. And, 'being happy' during those endeavors gives us a 3% bonus for effort.

7. Logic

a. M T W T F S S – The first letter in the seven days of the week starting with Monday (M).

b. 5 4 5 + 5 = 550 – The first plus sign (+) becomes the number four (4) with a short stroke of the pen. [anything] Just write the word anything in the space.

8. Science

Answer: The pond’s water level falls. The explanation is based on Archimede’s Principle which states that the upward buoyant force exerted on a body (boulder) immersed in a fluid (water), whether fully or partially submerged, is equal to the weight of the fluid (water) that the body (boulder) displaces. In other words, when the boulder is in the boat, the volume of water displaced has a weight equal to the weight of the boulder. But when the boulder is submerged, it displaces a volume of water equal to the boulder’s volume. Therefore, quantity-wise, more water is displaced when the boulder is in the boat (ponds water level is higher) than when it is submerged and the water level is actually lower. This can be verified mathematically by using specific gravity values for water and the boulder.

9. Logic

From the answer you immediately know the correct path because the response from either brother will be the same – the incorrect path. Why? Let’s say you ask the truthful brother first. He must tell you that the lying brother says the incorrect path. On the other hand, if you ask the lying brother initially, he will lie about his truthful brother’s response.

10. Cards

Answer: It’s true that I removed a card – page 8 has 6 cards and page 9 has 5 cards. But the reason this trick works so well is because none of the five cards on page 9 are the same as the six cards on page 8. They just look similar. So, it makes no difference which card on page 8 that you concentrate on, it has to be missing on page 9.

11. Algebraic Math

Answer: X x Y = 0.5

Here’s how this problem is solved.

(1)

4X-Y=3Y+7

4X = 3Y+7+Y

4X = 4Y+7

Substituting Y (2) in (1),

4X = 4 [0. 5 – X/8] + 7

4X = 2 – 0.5X + 7

4X = 9-0.5X X = 9/4-X/8 X + X/8 = 9/4 8X + X = 72/4

9X= 18

X=2

Substituting X = 2 in Y equation (2),

Y = 0.5 – x/8 and, Y = 0.25

Therefore, Xx Y = 2 × 0.25 = 0.5

12. Numbers/Math

Answer: The answer to this problem is 2. In solving the problem, ignore the first two lines of numbers because there are no instructions to do anything with them. Those who chose to include them in their P-S process simply got incorrect answers. Even some who limited their P-S to the third line incorrectly believed the answer to be 1 - the logic being that 1+1=2 and 2XO=O, therefore 0+1=1. The key to finding the correct answer is making l X O a separate operation in the equation as follows: 1+(l X O) +1= 1+0+1=2.

13. Logic

Answer: Here are the steps to solve this problem:

1-a. If Teddy is telling the truth, Sabrina’s two statements are false and she is a Knave. If, on the other hand, Teddy is lying, Sabrina’s statements are true and she is a Knight. In either case, she is not a Knark.

1-b. This analysis also applies to Virgil so that Ursula is not a Knark.

1-c. The same analysis applies to Wilma and therefore Virgil is not a Knark.

2-a. But Teddy said in his statement #4: Virgil is a Knark. Therefore, Teddy is confirmed to be a Knave.

3-a. If Teddy is a Knave, then his two statements are false and Sabrina is the truthful native, the Knight. Also, Wilma is a Knark.

4-a. Ursula is a Knave because both of her statement s are now shown to be false.

5-a. Finally, Virgil is a Knave because both of his statements are now known to be false.

Just to repeat from 3-a, Sabrina is the truthful native (Knight) and she is the one I approach for my supplies.

14. Math

Answer: The daughter is 44 years old. Here’s the solution:

1. Let the daughter’s age be X

2. The son is Y

3. And, the father is Z

The three equations we must solve are: X = Z/2, V + 22 = Z/2 and X = 2Y

Substituting Z=2X in Z = 2Y + 44, we get X = (2Y + 44)/2 and X = Y + 22. Substituting Y = X/2 in X = Y + 22, we finally get 2X = X + 44 and X = 44. So, the daughter is 44 years old and it follows that the son is 22 and the father is 88.

15. Numbers/Logic

Answer: The last two numbers to this sequence are 54 and 58. These values are determined by applying the following sequential mathematical operations: X2, +1, X2, +2, X2, +3, X2, +4

16. Numbers/Math

1+4=5

2 + 5 = 12

3 + 6 + 21

4 + 7 = 32

5 + 8 = 45

6 + 9 = 60

7 +10 = 77

8 +11 = 96

The numbers in the first two vertical columns increase by one progressing down the columns while the difference between numbers in the third column increases by two: 7+2 = 9 + 2 = 11 + 2 = 13, etc.

17. Infinity/Numbers

Answer: I believe the number sequence increases endlessly. The largest number I am familiar with is tens of trillions.

Perhaps this number comes to mind after frequently hearing about this country’s debt (\$) and the cost of conducting war during the past 14 years.

My answers bring me to the subject of ‘Limits’. To say that the number sequence increases endlessly has no practical meaning in our ‘finite’ world. Therefore, we assign a limit (a maximum number)- something that we understand and provides familiarity. For some circumstances we don't have to assign limits because they already exist. As an example, take a square piece of paper and fold it in half. Fold it in half again and keep repeating this process. From a mathematical perspective we acknowledge that the paper can be folded an endless number of times – 1, ‘J/2, l/4, l/8, l/16 …….. . However, in reality, the paper thickness becomes a limit and consequently no more than seven equal folds (:1/128) are possible.

Other limits will be considered for the problems that follow.

18. Infinity/Numbers

Answer: I believe 0.33 to be a reasonable decimal equivalent to the fraction 1/3. To get this result, we apply a practical limit to the seemingly endless sequence 0.333333……. The question is do we make the limit 0.333, 0.33 or 0.3 ? Actually, any one of these three may be acceptable but 0.33 is generally considered the best answer. This is an example of limitation based on practicality and consensus of opinion.

19. Infinity (Pi)

Answer: The area of a circle whose diameter is 50 inches is:

Area= Pix (diameter/2) squared= 3.14 x {50/2) squared= 3.14 × 25 squared = 3.14 × 625 = 1,962.5 square inches. My answer is an approximation because I used a limited value of Pi – one most commonly used in mathematics. The use of larger values like 3.14159 with an answer of 1,963.49, or an even larger value of 3.14159265358979238 (answer of 1,963.5) does not change the original answer significantly. Therefore, for reasons of brevity {there is no way I can deal with a number having endless decimal places) I limit Pi to two decimal places.

20. Infinity/Math/Physics

Answer: Both mathematics and empirical testing confirm that short wavelength radiation like x-, gamma-, and cosmic- rays cannot be totally stopped. Looking at mathematics first, an equation that best describes this phenomenon is the following classic radiation absorption law:

[where I is the radiation intensity emitted after passing through the subject, l{o) is the initial radiation intensity entering the subject, e (from the “God’s equation”) is Euler’s number (2.71828183), )J is the radiation absorption coefficient for the material the subject is comprised of, and x is the subject thickness].

Basically, this equation reveals how the initial radiation beam intensity [l°] is reduced to intensity (I) by partial absorption in a specified subject (material) of thickness x. With increasing thickness x, intensity (I) decreases endlessly towards ‘O’ but never reaching it. The ratio of 1/l° X 100 gives the percent reduction of l°.

From an empirical perspective, the following experiment demonstrates the practical use of the absorption equation. With a radiation detector (Roentgen meter) located four feet from an x-ray generator, make a 200 kV (kilovolt), 10 milliampere-minute exposure. Measure the radiation [l°] at the detector to be, say, 10 Roentgens (10 R ). Next, make a series of 10 milliampere -minute exposures using increasing thicknesses of lead sheets (lead is known to be an excellent x ray absorbing medium) located between the x-ray generator and the detector but close to the detector. A table listing the different lead thicknesses, the resulting radiation intensity (I) values in Roentgens (R) and the ratio (percent) of intensity (I) to the initial intensity [l°] is presented below.

These experimental results show that no matter how much lead is between x-ray generator and detector, the emerging radiation intensity is continuous albeit eventually diminishing to a level that cannot be recorded. State of the art radiation detectors simply aren’t sensitive enough. Also, we find that at some point the radiation emerging from the lead is small enough to satisfy the ‘safe’ levels required at the dentist office, a hospital x-ray room or an x-ray facility inspecting jet engine turbine blades.

The conclusion with this discussion is that the original x-ray beam wasn’t stopped – it was reduced to a safe level. The medium used to make x-radiation safe is usually lead. In the case of gamma radiation, safety is assured by appropriate thicknesses of concrete.

What we’ve learned from these different examples is that when it comes to ‘endless’ and ‘infinity’, there is a distinction between mathematics and the real world. Is there something we can learn from this distinction?

21. Universe/Astronomy – Cosmology

This question, like so many others – is there a ‘creator’?, is the earth round (not flat)?, and does the earth circle the sun (not the reverse)? – can be philosophical in nature. It all reduces to what we believe to be true. We believe the earth is round and that it circles the sun because scientific evidence tells us so.

But, it is a belief and I happen to be one of a vast majority who believe that strong scientific proof makes my belief factual. On the other hand, there are a few who continue to believe the earth to be flat. Those individuals belonging to the 'Flat Earth Society' believe only what their senses tell them and because they did not personally participate in any scientific projects or activity that demonstrates the earth to be round, they feel there is no reason to believe otherwise. However, the question about the existence of a creator is quite different. There is no scientific foundation for either supporting or refuting the existence of a creator. Therefore, as I see it, a belief in a creator is just as valid as the opposite belief. The size and age of the universe falls in the same category. There is no scientific evidence to lead one to believe the universe has a specific size - and is not infinitely large. Likewise, there is no evidence to believe the universe's age is limited to 13.7 billion years, or the opposite -that it has always existed. So, it simply boils down to what one ' believes.' The knowledge I've gleamed from a life-long study of astronomy & cosmology beginning with my early experiences in college mathematics and culminating in recent revelations by leading astrophysicists leads me to believe the universe is infinitely large and has existed an endless period of time. WHAT DO YOU BELIEVE?

Special Notation re Problems# 1 & # 4: As I described in #1 & #4 Problem Statements, I encountered these two problems in the sixth grade and they helped me understand what P-S is about and how to use it to my advantage. By presenting such ‘trick’ questions, my teacher demonstrated why one must be careful about quick (not thought out) solutions to a problem. They are often wrong because P-S techniques were not always fully utilized!

During his fort y-year career working in nondestructive testing, Roy Wysnewski authored numerous technical articles about industrial radiography some of which were published in Materials Evaluation (M.E.), the monthly technical journal for the American Society for Nondestructive Testing. Since his retirement in 2000, he published two additional articles (M.E.) and one e-book about industrial radiography.

In recent years Roy directed his writing skills to several non-technical subject s. The first, an autobiography entitled "My Boyhood Years", was written in 2014 but never published - it was dedicated exclusively to family. In 2015, an e-book about the authors life-long interest in, and experiences with, the weather - "Whether To, Or Weather Not To" was published. And, most recently Roy wrote about another life-long subject of close personal interest - problem-solving. This was just published as an e-book entitled "Having Fun With Problem -Solving".

Roy lives in Sarasota, Florida with his wife Judy and can be reached at his e-mail address: [email protected].

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