Engineering Quarterly Summer 2021

Engineering Quarterly VOL.III. . . No.4

SUMMER 2021

W ELCOME !

The Engineering Quarterly is a newsletter published four times each year by the University of Mary School of Engineering.

In each issue, we try to include articles from each of the five engineering majors that we currently offer on campus: Electrical Engineering, Mechanical Engineering, Civil Engineering, Computer Science, and Construction Management. The intended audience of the Engineering Quarterly is engineers, engineering students, engineering alumni, future engineering students, and people working in fields related to engineering such as science, computing, and mathematics.

In this issue we describe our summer activities. We are building a new building, teaching summer courses, performing service activities, and preparing for the fall. As usual, we also include some problems from our courses that you might like to try. Enjoy!

Dr. Terry Pilling Dean of Engineering

T HE J OURNEY OF AN O RDAINED E NGINEER

In 1991, a Catholic priest from the Latin Patriarchate of Jerusalem rode his car and drove for three hours in the Holy Land to visit a class of 7th graders at College De La Salle (Fréres), a Catholic French School in the heart of Amman, the capital of Jordan. Among these 7th graders was Khalid, who was all ears for what the priest had to say. The Catholic Father came to promote a boarding school in Beit Jala, a small city in the Holy Land. There, the

students would finish their middle and high school education, and later would be led into seminary, where they would finish their theological studies, and eventually become priests.

The priest encouraged all interested students to return to their parents that day and discuss this proposal with them. Khalid was fascinated and got very excited. He could not wait to return home and discuss this with his parents. This was the first topic that he opened with his mom during lunch.

“NO”, was the answer from his mother, who threatened to tell his father, who was all against any type of studies that does not put food on the table. His father had high hopes for Khalid, who was at the top of his class, he wanted him to become either a doctor or an engineer, as most parents dreamed that their children become in the Middle East region of the world.

“But why?” asked Khalid with a tone of disappointment mixed with the tears that were moistening his cheeks. “We need you to be successful” quipped his mother, “And we want you to get married and have children, I want to be a grandma one day” ... Khalid was utterly saddened. He tried to converse with them for the next few days, but it was a lost cause.

This was the first big “NO” in Khalid’s self-plan for his life. For sure, it was not the last.

Khalid had no choice but to finish his middle school at De La Salle. His high school was completed at one of the high schools that were newly established by the Queen of Jordan for gifted students. During High School, there were some American teachers, and many students there were fortunate and well to do.

They would visit the US in the summer and return to tell their friends unlimited stories about the beauty, technology, and advancement of the US. It made sense to any high achiever that the US is the land of success. So was the case for Khalid.

In senior year, he got a scholarship to finish his undergraduate studies at Texas Tech University, his parents, as was the norm then, were not comfortable sending their 17-yearold to a country so far that they know nothing about, and know no relative or friend in that state to help him if he needed while he was there. The answer was another “NO”. This meant one thing, that he had to face the TExam.

The T- Exam (Tawjihi) is the dreaded National High School Exam that was a requirement for all high school students in Jordan, the grade of which was the basis for the extremely competitive university admission. It was the nightmare that all families had to face at one point in their children’s lives. At that time, there were only five public universities in Jordan. School admission was not based on the students’ choice, but rather on their Tgrade. Top students went to the limited seats of Medical School, then the seats of Dentistry opened, then Pharmacy, then Engineer-

University of Mary

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ing, and so on. Khalid was able to secure a grade that would allow him to get into the engineering school at the University of Jordan, where he studied Electrical Engineering for five years until he finished in 2002.

He received a master’s degree in Computer Engineering from the University of Massachusetts-Lowell in 2004, and a PhD in in Electrical/Biomedical Engineering from The George Washington University in 2011.

While that would look impressive on a resumé or Linked-In profile, there was more to the actual story that was less glamorous. They say that the devil is in the details. In reality, it was God who was governing all the details.

During his studies, God was always in the picture. The effect of God’s presence was not only growing, but opposing many of the paths and plans that Khalid had for himself.

As any graduate student who studies in the US, Khalid was always looking for a scholarship to finish his studies. Unfortunately, both Universities where he did his graduate studies did not offer scholarships for international students at the School of Engineering. Furthermore, international students were not able to work full time in the US. Khalid was unable to support himself because the law required him to be a full-time student. His part-time work on campus would only pay his phone and electric bills. This was another huge “NO” that left him financially dependent on his parents until he finished school at the age of 30. Along with a change of his thesis project by his advisor that caused him to lose two years of work and start from scratch, Khalid went into medical depression. It lasted for a year.

However, God Who is all Love, and Who is beyond Love, does not leave his children stranded in life without support and without His Grace. Through all the disappointments in Khalid’s education, God arranged for Khalid to meet his future wife Sara, a native Jordanian and the daughter of a Greek Orthodox Priest, through a miraculous mutual connection in Jordan, while they were both studying in the States. The mutual connection was a man whose name literally meant “Grace”.

Khalid was hoping that upon graduation, he would work in a big company that would make the huge amount of investment that Khalid’s family put lovingly from their heart and their own pocket all worthwhile. He applied to hundreds of posts in DC. Most of them required security clearance, which was impossible to get as an international student.

He also applied to many posts across the country, but the US was still suffering from recession. His work in the Engineering field was not happening in the US. As he was closer to graduation, and being in the nation’s capital, he was able to get connected to two individuals who were at the highest rank of the World Bank Group, that promised to help him get employed there. This also was to no avail.

His dreams and his own definition of success were slowly fading away ... God had a totally

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different definition. The outcome of those investments would not be financial, but rather spiritual.

The light of the end of the tunnel became a little apparent, Khalid was able to work as an academic advisor at the UAE embassy in DC, while his wife was working at the Jordanian Embassy. Less than a year later, the couple moved with their two children to their home country of Jordan. They were hoping to have a better chance to serve the church as working professionals while being close to their families. Khalid was appointed as an assistant professor at the first Catholic University in Jordan, The American University of Madaba, which was owned by the Latin Patriarchate of Jerusalem. The past road that was filled with “NO” signs, started to slowly evolve to showing the “YES” sign, and both signs were directing to the same destination.

In 2013, Khalid and a friend connected with the directors of the Orthodox Educational Society which is the main umbrella for several Orthodox Schools in Jordan. They founded a “Focus Group” and started planning many activities with the sole purpose of bringing the Orthodox Christians in Jordan closer to the Church and to each other.

They were able to plan many activities, among which visits to different churches across Jordan, tourism trips to religious and biblical sites, soccer tournament for all the fellowships across the country, and lectures and Q and A sessions in parishioners’ homes by priests.

Khalid also received the blessing from Sts. Constantine and Helen parish priest in Amman to establish a fellowship ministry for college and working professionals. He was responsible of giving them the weekly lectures and devotionals as well as organizing activities with them. He was also a committed participant in the Divine Liturgy and church services as a member of the choir. Khalid has also completed three years of studying Byzantine music notation and chanting.

It became clear that Khalid’s career was not destined solely for academia or engineering. The calling to serve the Church was clear as the sun. Khalid took the blessing from his spiritual father to pursue ordination to the priesthood. His spiritual father was happy to hear the news, gave him the blessing, but urged him to study Theology first. In 2016, he enrolled in the Antiochian House of Study, a theology school in New Jersey and California that offered a program by correspondence with certain residency requirements. He received his Master of Theology degree in 2020.

Khalid was invited to serve as the office director for the newly elected Orthodox Bishop of Jordan, to help with the reorganization of the Archdiocese. While he was serving there, a faculty position opened at the University of Mary, Khalid took the chance and applied. He was hired and started working in Fall 2019.

In His words, Khalid would say: “I knew that the University of Mary would be great, but when I came here, it was beyond my farthest imagination. This is a place like no other!” He adds: “Here, your whole life can be formed to become a vocation, not just a career. Here, both students and faculty are formed to fulfill their calling by God, and the

Engineering Quarterly SUMMER 2021 reason of their life on earth, no matter what they do in life” Khalid and was ordained in the Antiochian Orthodox Church in April 2021 as Deacon Damian. The president of the University, Monsignor James P. Shea, the vice president, Diane Fladeland and her husband, as well as several of his Engineering colleagues were present at his ordination.

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These presentations are being developed together with hands on exercises for possible deployment as workforce development opportunities and cover the same content that will be presented in EME 298, our course on manufacturing technology. The topics covered included shop safety, metrology, machining, and welding. We in the Engineering School are excited to have Leland joining us this fall!

R ECORD E NROLLMENT The University of Mary has a record number of new freshmen students coming in the fall and that record includes a record number of new Engineering students with it. We have We at the University of Mary School of En- 64 new freshmen joining us in the fall which gineering are blessed to have Damian as our is divided into disciplines as follows friend and colleague and we fully support his Department New Total plan to become a priest and start an Orthodox Civil Engineering 13 38 church in the Bismarck/Mandan area while Electrical Engineering 9 27 he continues to design and teach courses and Mechanical Engineering 23 67 guide students as a professor in our departComputer Science 11 30 ment of Electrical Engineering.

Construction Management 8 27 Total 64 189

N EW M ACHINIST J OINS OUR T EAM

By JAMES CARRICO

Leland Vetter, an experienced machinist with many years of both teaching and industry experience, is joining us this fall as our new Shop Czar. Leland will teach safety and the use of the machine shop equipment as well as overseeing and guiding the activities of our students using our machine shop. Leland joined us in July and presented a series of topics on machine shop safety and fundamentals to our Mechanical Engineering faculty.

We have split several of our freshman courses into multiple sections to accommodate these new students since we have a policy of maintaining small class sizes. It is fundamental to our program that we have no more than 20 students in the majority of our courses.

We believe that it is much easier for students to learn when the class sizes are small since this enables more one-on-one interaction between the professor and the student, encourages active participation in class, student presentations, and allows us to combine some of our theory courses with their corresponding laboratory courses. Combining theoretical study and laboratory experience enables the student to immediately apply their technical knowledge to practical application, experimentation, and testing. This is crucial to solidifying the understanding of the abstract mathematics and technical material by putting it in a practical context. As our enrollment continues to increase we will continue to calve off multiple sections of courses until we reach our overall enrollment cap of 250 students.

We have designed our building so that the classrooms and laboratories are optimized for 250 students and our ultimate goal is to have that number as a steady state.

N EW C ONSTRUCTION M ATERIALS L ABORATORY

This summer we began construction of our new addition for our Construction Management and Civil Engineering programs.

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Engineering Quarterly SUMMER 2021 MUS 212 Concert Choir (3 semesters) GLOBAL STEWARDSHIP:

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booth there for the University of Mary and the School of Engineering.

POL 101 Responsible Citizenship and one of the following: ANT 171 Cultural Anthropology SOC 107 General Sociology HIS 103/104 Western Civilization and World History I or II ECN 101 Economics of Social Issues CTH/HIS 330 Catholicism & The Modern World CLA/HIS 311 Grandeur of Rome (Rome campus) Language (any 300 level or above) THEOLOGY:

The Construction Materials Laboratory was sponsored by Miller Insulation and will contain our wet labs for concrete and asphalt work as well as a crane and testing equipment.

THE 120 Search for God THE/CTH 234 Benedict - yesterday & today PHILOSOPHY: PHI 108 Search for Truth and one of the following:

Our main purpose in being at this conference is to raise awareness among the companies about our engineering program and our students. Hopefully the contacts that we have made will be a source of internships and possibly careers for our students and graduates.

PHI 208/308 Philosophical Ethics PHI/CTH 210 Search for Happiness: Faith and Reason in Life.

SENIOR ASSESSMENT: HUM 499 senior assessment

Construction Management Degree Requirements

Our Civil Engineering and Construction Management departments are looking forward to the completion of this new addition to our building which is scheduled for fall of 2022.

Up until this point we have been lucky to have local companies allowing us to use their laboratory and testing facilities to teach our students. Having our own facilities attached to our engineering school will make the logistics much easier for the students and it will also allow us to use the facilities and equipment at our discretion which is necessary if we want students to be able to work on projects around the clock as we do with our other laboratories in the Engineering building.

N EW C ONSTRUCTION M ANAGEMENT P ROGRAM We have completed the design of our new Construction Management degree which has been in development since spring of 2019.

Professors Matthew Scerbak and Richard Garman have used their industry experience as well as the mission and values of the University of Mary to design an outstanding program which is unique in the nation.

Common Core Requirements (taken by all majors) FYE 122 First Year Experience Seminar (with President’s Seminar) ENG 121 Composition II COM 110 Oral Communication

ART 127 Introduction to Drawing BUS 215 Principles of Management BUS 454 Project Management CHE 111 General Inorganic Chemistry I CHE 111L General Inorganic Chemistry Lab CMT 257 CM Fundamentals CMT 321 Construction Materials and Methods I CMT 322 Construction Detailing and Analysis I CMT 343 Construction Cost Estimating I CMT 345 Bldg Mech and Elec Systems CMT 360 International Building Code CMT 421 Constr Mat and Methods II CMT 422 Constr Detailing and Analysis II CMT 430 Construction Safety CMT 443 Construction Conceptual Estimating CMT 466 CAPM Certification Seminar CMT 470 Constr Scheduling and Control CMT 481 Senior Capstone I CMT 482 Senior Capstone II COM 414 Business Communication ECI 313 General Surveying with Lab ECI 313L General Surveying Lab ECI 444 Contracts & Specifications ENR 101 Introduction to Engineering ENR 200 Computing for Math & Engineering ENR 201 Engineering Mechanics I ENR 203 Engineering Mechanics II ENR 203L Engineering Mechanics II Lab ENR 460 Engineering Economy GLG 203 Earth Dynamics & Geophysics MAT 209 Calculus with Analytic Geometry I PHY 203 Introduction to Physics PHY 203L Introduction to Physics Lab HUM 499 Senior Competency Testing Choose Two Concentration Options: Business Leadership BUS 326 Management/Leadership Concepts BUS 328 Organizational Behavior Business Management ACC 101 Principles of Accounting I BUS 401 Operations Management Engineering Systems ENR 210 Computer Aided Measurement ENR 419 Engineering Data Analysis Soil Mechanics ECI 412 Soil Mechanics with Lab ECI 414 Foundation Engineering Structural Mechanics ECI 351 Structural Mechanics with Lab ECI 451 Steel Design ENR 445 Reinforced Concrete Additional Required Courses: ENR 470 is waived for students who select the Structural Mechanics concentration.

CMT 411 Interdisciplinary Fluency ENR 470 Engineering Ethics Total: 135-137 Semester Credits Program length: 4 academic years/128 weeks

ART: SELECT ONE OF THE FOLLOWING ART 108 Introduction to Photography ART 116 Introduction to Film ART 121 Art of Rome and Paris (Rome campus) ART 127 Introduction to Drawing ART 210 Topics in Textile Arts COM 200 Visual Communication ENG 140 Introduction to Narrative THR 105 Acting Technique I MUS 196 Introduction to Music MUS 211 Concert Band (3 semesters)

W ILLISTON BASIN P ETROLEUM C ONFERENCE

R ISING L EADERS ACADEMY V ISITS C AMPUS

By RODRIGO DA COSTA APARECIDO

On June 21-23, the University of Mary hosted the 2021 Rising Leadership North Dakota for high school students entering their freshman through senior year.

Students who take part discover: • Keynote topics of leadership, motivation, and attitude.

• Tours of various businesses in the Bismarck-Mandan community.

• A career fair and networking opportunities

The Williston Basin Petroleum ConferThe event included visits to the different ence was hosted in the Bismarck Event Cen- academic schools on campus and visits to area ter from May 11 to May 13, 2021. We had a businesses.

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Engineering Quarterly SUMMER 2021

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The Go Baby Go project here at the University of Mary was initiated five years ago as a partnership between Physical Therapy and Engineering where Engineering faculty and students work side-by-side with Physical Therapy doctoral students to adapt ride-on toys that help with the inclusion and development of children with special needs.

When the students visited the Engineering School we first told them a bit about our engineering program and the types of careers someone can get with an engineering degree.

Then we gave each of them electronics breadboards, wires, batteries, and LED’s and we challenged them to wire several circuits to light up the LED’s.

gineering Mechanics II (Deformable materials) + Lab, as well as the University’s algebrabased physics course PHY 203 - Introduction to Physics.

All of these courses were accelerated 7week courses. This was the first summer that the entire Engineering Mechanics course sequence was offered in one summer and that either course was offered in a 7-week format.

Just like in the normal academic year, students were tasked to do hands-on assignments in addition to the traditional textThe PT students would decide how to best book course material. Students in ENR 201 accommodate the needs of the child and the (in-person and online) successfully designed engineering students would design the frame and built catapults that launched a racquetand electronics to meet those needs. ball to hit their professor in the face (or their target) 10 meters away.

The intent was to show them how electricity works and also how you can wire circuits with breadboards.

The students worked over several weeks The event went well and the students seemed to enjoy the challenge of building to design the modifications that would benefit electric circuits. A show of hands revealed the children and their families. several of them plan to go to college for enThey added elecgineering after they graduate. tronic features and modifications to the frame, along with G O BABY G O ! special compartBy RODRIGO DA COSTA APARECIDO ments to hold backpacks and feeding equipment when necIn 2006, Cole Galloway started an incluessary, and after a sion movement for children with motor imfew weeks of design pairments, called Go Baby Go, at the Uniand build the students versity of Delaware’s Department of Physihad the great pleacal Therapy. This program includes modifysure of giving the ing an electric toy car in any way necessary cars to the children for a child with motor or developmental delays to access their environment and interact and watching them learn to use the controls to move on their own. It was a beautiful experiwith their peers. ence.

S UMMER C OURSES

By ERIC GARCIA

The engineering building housed three courses this summer. These were the sophomore level engineering mechanics course series: ENR 201 - Engineering Mechanics (Particles and Rigid Bodies) and ENR 203+L - En-

Students in ENR 203L successfully designed and built a paper truss bridge and 3D printed cable bridge to meet certain weight and deflection requirements. The paper truss pictured spanned a 2 foot gap, weighed less than three pounds, and carried 67.5 lbs before failing!

In addition to the Engineering Mechanics courses, Physics 203 was offered this summer in the Engineering facilities and might have been the first class of students to be hosted in the engineering building that was all women. Both the lecture and lab sections were held in the Engineering facilities. Students in Physics 203 even used some of the catapults designed by students in ENR 201 in past school years to analyze projectiles and perform calculations to determine where their

VOL.III. . . No.4

professor should stand to hit him in the face.

The pictured team succeeded. It was not a good summer for Dr. E. Garcia’s face.

Engineering Quarterly SUMMER 2021

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(c) ΣFx = 0 T − fs = 0 348N − µs (84kg)(9.81m/s2 ) = 0 µs = 0.42

• PHY 203 Problem: (a) 11m (b) 0.61m/s2 (c) 3.67m/s

ENR 201 P ROBLEM

By ERIC GARCIA

My two-year-old daughter got a head start. • ENR 203 Problem: She was running at a constant velocity of 1.5m/s and was 2m ahead when my nephew FA = 628.63kips began to run. My nephew started at rest and accelerated at a constant rate until he caught FB = 586.67kips up to my daughter 6 seconds after he started FC = 544.70kips running.

In order to steal a computer file from a (a) How far did my nephew run when he caught up to my daughter? secured vault, a spy and an accomplice must C ALCULATING e drop down through an air vent in the ceilBy TERRY PILLING ing to access the computer while the vault (b) What was my nephew’s acceleration? is locked. The spy is lowered down to the (c) How fast was my nephew running when computer using a harness and pulley syshe caught up to my daughter?

One day during the summer I and several tem, which is anchored to the vent and by of the other faculty were in the office working the accomplice (as shown in the picture). on our ABET accreditation rubrics and our lecture notes for our fall courses when I stumENR 203 P ROBLEM bled across an interesting problem from the

By ERIC GARCIA internet. I mentioned it to the others and we ended up spending a couple of hours chatting The rigid beam shown is pinned to about it and working out the solution. Supthe top of the three square 1.5f t × pose you start selecting random numbers be1.5f t columns made of concrete that tween 0 and 1 and adding them together. On has a compressive strength of 3500psi. average, how many selections will you have to make before the sum exceeds 1? For example, suppose my first pick is 0.2 and then my second pick is 0.9. The sum is 0.2 + 0.9 = 1.1 and since it is already bigger than 1 I stop after making only two selections. If I did it again Once the spy has accessed and downloaded I may get 0.13 + 0.54 + 0.21 + 0.89 = 1.77 the necessary files, the accomplice pulls on where I only exceeded 1 after the fourth pick. the rope and raises the spy back to the vent.

Notice that in the first case I only went to Assume the mass of the rope and harness is one decimal place and in the second I went to negligible, there is no friction in the pulley, two decimal places. We are assuming an infiand the rope is always completely taught. nite number of decimal places in our choices so that in each of the above examples we are (a) Determine the tension force in the rope assuming an infinite string of zeros after the if the spy has a mass of 67kg and is Determine the force supported by each col- digits I have shown. lowered at a constant acceleration of umn if the rigid beam is subjected to the So in my two example trials I first exdistributed load shown. Assume the calcu0.4m/s2 ceeded 1 after only two selections and then lated forces and deformations are located at I exceeded 1 after 4 selections. If we stop at (b) Determine the pulling force the accom- the center of each square column. only two trials then the average number of seplice must exert to raise the spy back to lections it takes to exceed 1 would be: the ceiling vent at 0.6m/s2 2+4 S OLUTIONS TO P ROBLEMS =3 Avg = (c) If the accomplice has a mass of 84kg, 2 what is the required coefficient of static In other words for my simple experiment with friction between the accomplices cloth• ENR 201 Problem: two trials it takes, on average, three selections ing and the vent to not slip when trying (a) of numbers between 0 and 1 before their sum to pull the spy up with the force found exceeds 1. in part (c)? ΣFy = may Now that you get the idea let’s see what 2T − W = may happens if we increase the number of trials. In 2T − (67kg)(9.81m/s2 ) = (67kg)(−0.4m/s2 ) order to do a large number of trials the easiest method is to just write a computer program to T = 315N PHY 203 P ROBLEM do it. Here is a program written in C where (b)

By ERIC GARCIA the user enters the number of trials to perform and then the computer will do them and, when ΣFy = may it is finished, it will output the resulting averMy two-year-old daughter and three-year- 2T − W = may age number of selections to exceed 1 among 2 2 old nephew decided to have a foot race down 2T − (67kg)(9.81m/s ) = (67kg)(0.6m/s ) all the trials. Here is the program: a long driveway.

T = 348N

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e.c # i n c l u d e < s t d i o . h> # i n c l u d e < s t d l i b . h> # i n c l u d e <math . h> # i n c l u d e < t i m e . h> # d e f i n e NUM 20 i n t main ( i n t a r g c , char ** a r g v ) { i f ( a r g c != 2 ) { f p r i n t f ( s t d e r r , " Usage %s TRIALS \ n " , argv [ 0 ] ) ; f p r i n t f ( s t d e r r , " TRIALS = number o f t r i a l s to average \ n" ) ; f p r i n t f ( s t d e r r , " The c o m p u t e r p i c k s random numbers \ n " ) ; f p r i n t f ( s t d e r r , " b e t w e e n 0 and 1 u n t i l t h e sum i s g r e a t e r t h a n 1 . \ n" ) ; return 1; } d o u b l e S , sum = 0 ; i n t TRIALS = a t o i ( a r g v [ 1 ] ) ; int i t e r a t i o n ; i n t h i s t o g r a m [NUM] ; l o n g MAX = pow ( 2 , 3 1 ) ; f o r ( i n t j = 0 ; j < NUM; j ++) h i s t o g r a m [ j ] = 0; srand ( time (0) ) ; f o r ( i n t i = 0 ; i < TRIALS ; i ++) { S = 0; i t e r a t i o n = 0; w h i l e ( S <= 1 ) { i t e r a t i o n ++; S=S+ r a n d ( ) / ( 1 . 0 * MAX) ; } i f ( i t e r a t i o n < NUM) h i s t o g r a m [ i t e r a t i o n ]++; } f o r ( i n t j = 0 ; j < NUM; j ++) sum=sum+ j * histogram [ j ] ; p r i n t f ( "%d %l f \ n " , TRIALS , sum / TRIALS ) ; return 0; }

If I compile the code using gcc e.c and then run the program for 1000 trials using ./a.out 1000, here is the output from the program: 1000 2.686000

Let’s write a Bash shell script which will start at a small number of trials (128 in the case below) and it will run the simulation and output the result. Then it will double the number of trials and run the simulation again. It will continue to double the number of trials over and over until it reaches a huge number of trials (229 in the case below) outputting the result each time. Then we will graph the resulting number versus the number of trials and see if, as the number of trials gets larger and larger, the result seems to converge on e.

Here is our Bash script: e.sh # ! / bin / bash i =128; MAX=$ ( echo " 2^29 " | bc −l ) i f [ −e e . d a t ] ; t h e n rm −f e . d a t fi w h i l e [ $ i −l e $MAX ] ; do l e t i = i *2; . / a . o u t $ i >> e . d a t done g n u p l o t −p e . g n u p l o t &

Notice that that it uses our C program to do each simulation and it outputs all of the results to a file called e.dat for later graphing and when it is finished it graphs the result using Gnuplot. Here is a Gnuplot script which will graph the results. e.gnuplot s e t t e r m x11 set t i t l e " I t e r a t i o n s to e" set xlabel " Trials " set logscale x 2 s e t format x " %.1 t x 1 0 ^{%T} " s e t format y " %.3 f " p l o t " e . d a t " u 1:2 with l i n e s , 2.7182818 w l

which means that in 1000 trials the average number of selections the computer had to make so that the sum was bigger than 1 was 2.686 selections.

Now what if I increase the number of tri- You might want to type in the C program, the als further? How about 5000 trials? Here is Bash script, and the Gnuplot script on your Linux command terminal and play with them. what I get: When I run the Bash script here is the graph that pops up: 5000 2.713200 and for 10,000 trials I get: 10000 2.713100 This is very interesting. It seems like, as I increase the number of trials, the result is getting closer and closer to 2.71 and this is a very interesting number. There is a mathematical constant which is very familiar to all calculus students called Euler’s Number, or e, which is the base of the natural logarithm and is almost as famous as π. Like π, e is irrational and so it , has infinitely many decimal places. The first 20 digits are: Notice that I have indicated the exact value of e as a horizontal line on the graph. Indeed, e = 2.7182818284590452353 it looks from the graph that as I increase the Now, the result of our little random number number of iterations the resulting value apexperiment is a number that looks like it could proaches e!

So it seems that if you select random numbe e. This could just be a coincidence and so what can we do to test the possibility? To be bers between 0 and 1 it will take you, on averabsolutely certain one should devise a mathe- age, e picks before the sum exceeds 1. Why is matical proof to show that the result of our ex- that? There must be some reason! Well, that periment will approach e as the number of tri- is what mathematics is for! Now that you have als goes to infinity. However, before we spend confidence that it is true, you should go about a bunch of time and effort finding a mathemat- finding out why it is true. This is exactly what ical proof we should first try to give ourselves finding a mathematical proof means. Truths and Facts are fun to know, but the reason why a bit more confidence that it really is true.

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See The Men of Mathematics by Eric Temple Bell.

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a truth is true, and why a fact is a fact, is what science and mathematics are all about.

In math the reason why something is true is called a mathematical proof and in science the reason why something is a fact is called a scientific theory.

D ERANGED C ARDS

By TERRY PILLING

I can’t help but show you another cool find.

As most engineering students know, Leonard Euler (pronounced “oiler” the same as the Edmonton Oilers hockey team) was a mathematician who did a huge amount of mathematical work that is used throughout engineering. In fact, Euler was born in 1707, one year after Benjamin Franklin, and lived for 76 years. His parents wanted him to be a priest, but after registering in Theology at the University of Basel, Switzerland at the age of 14 he quickly discovered a love for mathematics and decided to switch majors. He would visit mathematician Johann Bernoulli in the evenings and work on interesting mathematics problems and he quickly became one of the greatest mathematicians of all time. He eventually took a professorship in Russia, got married, had 13 children, and produced so many papers and books that his Opera Omnia, or collected works, is still being compiled over 300 years later where, at present, it consists of 73 volumes with many more yet to appear. To say he was prolific is an understatement since he was the most prolific mathematician of all time. He was called “Analysis Incarnate” by mathematical biographer Eric Temple Bell1 .

The story I want to relate in this article begins with a card game that was played back in the 1600’s called ‘coincidence’ where a player bets that as the cards are dealt from a shuffled deck a coincidence will occur where a card will come out in the exact same spot that it was originally in the unshuffled deck. In other words, the dealer takes 13 cards from Ace to King, shuffles them, and then deals them out one at a time. As the cards are dealt the players count “one”, “two”, etc. and if, for example, they count “eight” at the same time the 8 is dealt then it is a coincidence and the dealer loses and player wins. If all 13 cards are dealt with no coincidence occurring the dealer wins and takes the players money.

Obviously, whenever there is money involved in any game, the player, and especially the dealer, want to figure out the odds of winning. So in the early 1700’s it was a bit of a challenge in the mathematical community to figure out the probability of a coincidence occurring. This is when Euler heard about it. He then proceeded to solve it and the result he got was a bit surprising. Let me show you how he did it.

The first thing he did is isolate what he was looking for. He thought about what had

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to happen in order for a coincidence to not occur. For no coincidence to occur, each of the shuffled cards has to reside in a different position from its original one. Euler knew that if you took 13 objects and you picked them at random, there are 13! (i.e. 13 factorial) different ways to do it. This is because there are 13 choices for your first pick, then 12 remaining for your second pick, etc. The total number of different arrangements is therefore 13! = 13× 12×11 · · ·×2×1 = 6227020800. That is a lot of different orderings! What Euler wanted to find out was how many of these had no cards that are in the same spot they started in before the deck was shuffled. In other words he wanted to know how many of the different arrangements were “derangements” – meaning no cards remained “arranged”. He invented the symbol Π(n) to denote the number of derangements of n objects. To get a feel for this let’s look at a couple of examples. Suppose you had only 1 object. Then obviously it will be the first one you pick since it is the only one you pick. Thus it will be in the same position it started in and hence it is not a derangement.

Hence there are no derangements of 1 object and Π(1) = 0. What about two objects, a and b. The original order is ab and if you choose them one at a time there are only two possible orders you can get. Either ab again or else ba.

The second case has neither a or b in the same spot it started in and so it is a derangement.

Hence Π(2) = 1. What about three objects with original order abc? Here are the 3! = 6 possible ways to pick them:

Engineering Quarterly SUMMER 2021 it’s a2 and then multiply by (n − 1) at the end to account for all the other choices. Since we also don’t want a1 in the second spot either, we need the remaining objects to be a derangement of a1 a3 a4 . . . an and there are Π(n − 1) of those.

Hence the total number of derangements of n objects is given by adding up the numbers we found in the two cases above Π(n) = (n − 1) [Π(n − 1) + Π(n − 2)] Euler found this recursion relation. Since we know that Π(1) = 0 and Π(2) = 1 we can use the formula to generate Π(3) Π(3) = (3 − 1) [Π(3 − 1) + Π(3 − 2)] = 2 [Π(2) + Π(1)] = 2 [1 + 0] = 2

7 Pi(13) = 2290792932 Probability = 0.367879

So the probability that the dealer will win is just over a third. So it looks like a pretty good bet for the player that one of the cards will come out in its correct spot!

This is not the end of the story however. If you take the inverse of the probability above you will find a very interesting number appears. Try it! Euler asked himself the following question: “what happens if n becomes really large?” Do things get better or worse for the dealer and by how much. Well it is easy to figure this out. Just start with his formula and take the limit as n → ∞. Let’s do it. The equation for the probability is

n Π(n) X (−1)k We can now happily write a computer proprobability(n) = = n! k! gram using this recursion relation and find out k=0 what Π(13) is. However, it would be more convenient to have a formula in terms of n and taking the limit gives rather than a recursion relation so we don’t ∞ X (−1)k have to compute all the previous ones in orprobability(∞) = der to get the next one. You can do this by k! k=0 working out a few of them from the recursion and trying to detect the pattern. Euler did this Now recall the power series representation the and found that exponential function n k X (−1) ∞ X Π(n) = n! xk x k! e = k=0 k! k=0 You can prove this formula by induction by knowing it is true for n = 1 and n − 2 We see that our probability equation is noth1 k ing but this series evaluated at x = −1. Hence X (−1) Π(1) = =1−1=0 k!

1 k=0 −1 probability(∞) = e = abc bca cab 2 e X (−1)k cba acb bac Π(2) = 2 = 2(1 − 1 + 1/2) = 1 k! and inserting the decimal value of Euler’s fak=0 of these, only the two in bold face are demous number e = 2.71828 . . . we find that rangements and so Π(3) = 2. Euler went on and showing, using our recursion relation, that for large n the probability that an arbitrary seto compute a few more and found Π(4) = 9, being true for Π(n − 2) and Π(n − 1) implies quence of cards will be a derangement is Π(5) = 44, Π(6) = 265, and so on. How- it is true for Π(n).

In any case, now we have a formula such ever, if you try to do this yourself you will probability(∞) = 0.367879 that, given n, we can easily find the number of find it excruciatingly tedious and pretty much impossible to do for the 13 cards we are af- derangements Π(n). We can write a simple C which is exactly what we got to six decimal ter. The number of permutations, 13!, is just program with a for loop to find that places already with n = 13. So clearly the setoo many to write down and then try to sift ries converges very quickly. It is remarkable Π(13) = 2290792932 out the ones that are derangements. So Euler Now we come to the probability. The that a combinatorics problem about the numneeded to figure out a formula instead. Here probability that you will get a derangement ber of derangements in dealing cards results is what he did: out of a randomly shuffled 13 cards – i.e. the in 1/e. It once again shows how these strange Start with n objects in order: a1 a2 . . . an probability that the dealer will win (since no irrational numbers like e and π tend to pop up and look at ways to derange them. First, in card is in its original spot) is given by the total in the most unexpected places. any derangement, a1 cannot appear in the first number of derangements divided by the total Read Euler, read Euler. He is the spot or it wouldn’t be a derangement. So there number of possible arrangements including master of us all. are two possibilities. Either a1 ends up in the derangements. We will include this com– Laplace the second spot, or it ends up somewhere else putation in our C program as follows down the line.

Challenge for the Reader Case 1: If a1 is in the second spot then one derange.c of the other n − 1 objects must be in Write a game which deals out 13 cards the first spot. Let’s just assume that a2 # i n c l u d e < s t d i o . h> and checks whether or not it is a de# i n c l u d e < s t d l i b . h> is there for now but multiply by n − 1 rangement. Then have the computer i n t main ( i n t a r g c , char ** a r g v ) { at the end to account for the same thing i f ( a r g c ! = 2 ) { p r i n t f ( " u s a g e : %s N \ n " , play against itself a few hundred thouargv [ 0 ] ) ; return 1;} with any of the others in that spot. Our sand times and see if you get the preint N = a t o i ( argv [ 1 ] ) ; int a = 1; sequence thus starts out a2 a1 and then dicted probability as calculated here. double k = 1 ; followed by a derangement of a3 . . . an . double n = 1 ; f o r ( i n t i = 1 ; i <=N ; i ++) { However, the number of derangements a = −1* a ; of a3 · · · an is Π(n − 2). And so in the n = n* i ; k = k + a/n; case where a1 ends up in the second } spot we have (n − 1)Π(n − 2) possible p r i n t f ( " P i (%d ) = %.0 l f , P r o b a b i l i t y = Contact Information %l f \ n " ,N, n * k , k ) ; derangements. return 0; For more information about our en} Case 2: In this case a1 does not end up in gineering program email: Engineerthe first or second spot. In this case ing@umary.edu one of the other (n − 1) objects are in Running this with n = 13 gives the following Next issue: November, 2021 the first spot (again we will just assume output

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