February 2002 - Mu Alpha Theta

Attend the 32 nd Annual National Convention in Mississippi!!
The Mathematical Log
Volume 46 Number 1 February 2002 ISSN 0025-5580
Mathematical Pathways You’ll Want to Explore
Dr. Don Allen
National Aboriginal Career Symposium
Ottawa, Canada
We shall be examining together a range of "number" and related topics that are featured in
Recreational Mathematics, the neglected "full” side of the subject. Within each topic we'll try to consider
just enough depth for you to grasp the notions--to "get the hang of it"--so that you can take it away, share
it with others, and investigate it further on your own.
Slicing a Pizza
Consider a pizza--a very ordinary one, flat and round. Cut it once--a single straight cut--and you get
two pieces. Cut twice and you get as many as four pieces--the pieces don't have to be equal, nor need
the cuts pass through the center. With three, four, five cuts, how many pieces should it be possible to
obtain Do you see a pattern in the numbers of pieces Can you predict the maximum number for nine
cuts Can you actually make a drawing that shows the maximum number
A related question extends to a third dimension. Find a round potato. Section it with plane cuts.
Reassemble the pieces after each cut. What is the maximum number of "chips" after two cuts, three cuts,
four cuts
A Bracelet of Numbers
The numbers that we deal with in this simple investigation are the ten digits, 0 to 9. Choose one such
number. Choose a second, which need not be different. Obtain a new number by adding the first two
numbers, and recording only the ones digit of the sum. Thus, for 8, 9, we have 8, 9, 7, ..., the start of an
unending sequence. Obtain a "next" number by similarly combining the final two. The sequence
becomes 8, 9, 7, 6, 3, 9, ...Continue. Predict what may happen, and investigate to see if you are right.
Try, at leisure, a triple of starting digits instead of a pair. How, otherwise, might the procedure be
modified to produce different, but interesting, results.
Hailstone Numbers
Known now for over 60 years, an attractive “number” investigation involves the so–called Hailstone
Numbers– you’ll also see references to the Collatz conjecture and to the "3x + 1" problem. Choose a
number-- any positive integer. The number that you choose necessarily is even or is odd. If it is even,
divide it by 2. If it is odd multiply it by 3, then add 1. You obtain in this way a new number. Repeat the
process with the number that you obtain. Continue to a sequence of positive whole numbers. Stop only
if you reach 1. Numbers that take you to 1 are hailstone numbers. We'll discuss why.
An example: You choose the starting number, 15. Your sequence is: 15, 46, 23, 70, 35, 106, 53,
160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1. Your sequence rises to a height of 160, and does reach 1-- in 17
steps; 15, therefore, is a hailstone number. Try 100 (25 steps). Try 31 (106 steps). Try 27.
Which numbers are hailstone numbers Are all numbers hailstone numbers What might happen if
you were to change the rules

February 2002 The Mathematical Log Page 2
Mathematical Pathways You’ll Want to Explore continued…
A Square of Differences
This simple number investigation can yield quite surprising results. Something seems to be
controlling what is happening, but it can be hard to see exactly what.
To begin, draw a large square. Then choose four numbers-- zero or positive whole numbers.
Assign each number, in order, to a corner of your square. Choose, for example, 3, 11, 2, 6. Now, add dot
at the midpoint of each side of your square. Join dots so as to make an inner "diamond"--actually, a tilted
square. At each corner of this square, write the number that is the difference (big minus little) of the two
numbers ending that side of the bigger square. Obtain, in the example we started with, 8, 9, 4, 3. Repeat
the process, getting 1, 5, 1, 5. Again: 4, 4, 4, 4. And again: 0, 0, 0, 0. How did this happen Will all
starting numbers take you to 0, 0, 0, 0 In so few steps Students once showed me numbers that
reached 0, 0, 0, 0 only after 15 steps. Can you equal that Also, what might have happened if we'd
chosen a triangle or a hexagon instead of a square
Reversals to a Palindrome
A "palindrome"-- a very old term-- is a number, word, or expression that reads the same left-toright
as right-to-left. As for examples, 383, 4664, and 1052501 are number palindromes. Similarly, eye,
level, and radar are word palindromes. Also, “Was it a cat I saw”, “Able was I ere I saw Elba” (attributed
to Napoleon), and “A man, a plan, a canal— Panama” are sentence palindromes.
We shall focus on number palindromes.
To begin, choose a number of two or more digits, probably not a palindrome. (If the number
happens to be a palindrome, then the activity is over.)
The “reversal” of such a number is the number obtained when the digits are written in reverse
order. Thus, the reversal of 326 is 623, and of 990 (let’s agree) is 099.
Write the number that you have chosen. Write its reversal. Add these two numbers. It’s quite
possible that, in one step (one reversal, one addition) you obtain a palindrome.
Thus, 623+326 = 949, a palindrome, in one step.
If not, repeat the whole process. Repeat using the sum that you obtained— writing the sum, its
reversal, and adding these two numbers. It’s quite possible that, in these two steps, you obtain a
palindrome. Thus, for the number 58: 58+ 85 = 143, then 143+ 341 = 484, a palindrome in two steps.
Further, 157 gives a palindrome in three steps:
157 + 751 = 908, 908 + 809 = 1717, 1717+7171 = 8888.
Explore.
Check whether 990 reaches a palindrome in five steps. Check whether 79 gives a palindrome in
six steps. Check whether 395 gives a palindrome in seven steps.
My students inform me that 89 gives a palindrome, but that some big numbers are reached and
surprisingly many steps are required.
Is every number either a palindrome or else a limited number of such steps from being one
Kaprekar’s Constant
This number activity has more general possibilities, but to introduce it we focus on its four-digit
case. First, write down a four-figure number (not all digits the same), say 3529. Now, rearrange the
digits to form the largest number that you can (9532, I suggest) and the smallest such number (2359).
Subtract these two numbers (large minus small), and get a new number, 7173. Repeat the process:
7731- 1377 = 6354. Repeat again: 6543 - 3456 = 3087. And again: 8730- 0378 = 8352. And again: 8532
- 2358 = 6174. Now, observe: 7641 -1467 = 6174… again, and again! For the amateur mathematician
who discovered it, D. R. Kaprekar (he lived in India), this number, 6174, is now known as Kaprekar's
constant. Four-figure numbers "lead to" 6174. Why Show that 9017 leads to 6174 in three steps, 9017
leads to 6174 in seven steps. Would there be a Kaprekar constant for three- figure numbers ...for fivefigure
numbers
What next A Math Fair project or math-related Science Fair investigation
could arise from such explorations. Some good reading I recommend
Martin Gardner’s many recreations books.
--- Dr. Don Allen

February 2002 The Mathematical Log Page 3
Win a trip to the 2002
National Convention!
What's at stake Convention registration for a sponsor and a non-senior student and travel expenses
up to $500 (travel arranged by the national office) for each of you to the national Mu Alpha Theta
Convention in Starkville, Mississippi. July 26-31, 2002. A total of twelve of these prizes will be
awarded— three per region.
Who is eligible Any newly chartered chapter or chapter that has not been represented at the national
within the past three years.
How to enter Write two paragraphs about your chapter, one describing the events your chapter has
participated in this academic year, and the other describing how your chapter will benefit from
representation at the convention. While you're at the convention, you will be asked to complete
surveys (one for the sponsor and one for the student) describing your feelings about the convention
and how your chapter will benefit from your participation at the convention.
Mail entries to:
Mu Alpha Theta National Office
601 Elm Ave, Rm 423
Norman, OK 73019
ENTRIES MUST BE POSTMARKED BY
MAY 15, 2002
News
• MAΘ National Convention • MAΘ National Listserv
Come for a heapin’ helpin’ of Southern hostipality, charm,
and…
`tzÇÉÄ|t `tà{
32 nd Annual Mu Alpha Theta National Convention
Mississippi State University
Starkville, Mississippi
July 26- 31,2002
Hosted by Mississippi School of Mathematics and Science
The Convention Registration Packet is now
available on-line at www.mualphatheta.org
For more information contact:
Claudia Carter (ccarter@msms.k12.ms.us)
or
Aubrey Knight (aknight@msms.k12.ms.us)
• MAΘ Website
Check out the National Office website at
http://www.mualphatheta.org/
for the latest information from the National Office.
In an organization as large and as geographically vast as Mu
Alpha Theta, a listserv can provide a convenient way for members
to keep in touch and up-to-date on events and topics. Since every
member sees each posted email, it provides an excellent way to
distribute information quickly. The list also provides a forum for
extended discussions. Members can read each post when
convenient and, if desired, submit their comments or suggestions.
There are two ways to subscribe:
1. Send an email to
MuAlphaTheta-subscribe@topica.com.
The listserv will automatically extract your email address and
add you to the list.
2. A subscription form is also available at:
http://www.topica.com/lists/MuAlphaTheta/subscribe/location=listinfo
If you still can't subscribe, send an email to Sam Koski at
ASKoski@aol.com with the email addresses that you want added.
• MAΘ Student Delegate/Chapter
President Listserve
Student Delegates and Chapter Presidents are invited to
join a listserv by sending a blank e-mail to mailto:subscribedelegate@topica.com.
Student delegates and chapter
presidents can all play a large role in the success of Mu
Alpha Theta this year, through this mailing list. Throughout
the year the listserv will provide a chance to have an open
discussion of any topics related to Mu Alpha Theta.
Sponsors are also invited to join the list to provide
additional suggestions and comments for members.

February 2002 The Mathematical Log Page 4
Mu Alpha Theta Awards
The Lucent Technologies Foundation is sponsoring an
annual international competition to identify and reward the
academic excellence and scientific achievement of 60
students. The Institute of International Education manages
the competition in the 18 participating countries/regions:
Australia, Brazil, Canada, China/Hong Kong, France,
Germany, India, Korea, Japan, the Netherlands, Saudi
Arabia, Singapore, Spain, Taiwan, the United Kingdom,
and the United States.
Chosen applicants receive $5,000 cash along
with an invitation to participate with Bell Labs researchers
and fellow Global Science Scholars from around the world
in the Global Summit during the summer of 2002. In
addition, the recipient will be considered for an internship
at Lucent Technologies in the home country.
Visit the Institute for International Education
website at http://www.iie.org/pgms/lucent/ for details,
including specific eligibility and application requirements,
a downloadable application form, and contact information
for the program coordinator nearest you.
Don’t forget!
The acclaimed Art of Problem Solving
series is still available from the
National Office. Check the website for
more information.
Misunderstanding
Each Mu Alpha Theta chapter would not be possible without the endless
hours of devotion given by its sponsor. To honor these dedicated
sponsors, the Mu Alpha Theta Governing Council presents the Huneke
Distinguished Sponsor Award and the Student Delegate Assembly gives
the Sister Scholastica Award. Sponsors who are recipients of these
awards must be present at the national convention.
Both of these honors recognize individuals who have instilled in Mu
Alpha Theta members a desire to explore the topics in mathematics that
are usually not covered in a high school mathematics curriculum.
In addition to honoring sponsors, the Mu Alpha Theta Governing Council
recognizes two members who are graduating high school seniors. The
Kalin Award, awarded at the National Convention, recognizes an
exceptional Mu Alpha Theta member. The Andree Mathematics
Education Award recognizes an outstanding student who is planning to
be a mathematics teacher. The Andree Awardee does not have to be
present at the National Convention.
If you would like more information on any of these award, visit
www.mualphatheta.org/awardships/awards.html. Please consider
nominating a student or a sponsor for one of these awards.
The Mathematical Log
Volume 46, Number 1, February 2002
The Mathematical Log is the official publication of Mu Alpha Theta, a
national high school and junior college mathematics honor society and
mathematics club federation. Founded in 1957 by Richard and Josephine
Andree, Mu Alpha Theta is co-sponsored by the Mathematical
Association of America (MAA), the National Council of Teachers of
Mathematics (NCTM), and the Society of Industrial and Applied
Mathematics (SIAM). Correspondence may be directed to Mu Alpha
Theta National Office, 610 Elm Ave., Room 423, Norman, OK 73019,
email: nationaloffice@mualphatheta.org, or to Log editor Pat Bowler
Johnson, New Trier High School, 385 Winnetka Avenue, Winnetka, IL
60093, email: bowlerjp@newtrier.k12.il.us.
© 2002 Mu Alpha Theta
National Officers:
President:
Harold Reiter
University of North Carolina, Charlotte, NC
Past-President:
Claudia Carter
Mississippi School for Math and Science
Secretary-Treasurer:
Paul Goodey
University of Oklahoma, Norman, OK
Governor Region I:
Tom Norris
Thomas Jefferson High School, Auburn, WA
Governor Region II:
Mattie White
Benjamin Franklin High School, New Orleans, LA
Governor Region III:
Susan Hiller
Vero Beach High School, Vero Beach, FL
Ah, you are a mathematician,
they say with admiration
or scorn. Governor Region IV: Gary Walker
Then, they say,
I could use you
Carter High School, Strawberry Plains, TN
MAA Representative: J. Douglas Faires
Youngstown State University, Youngstown, OH
NCTM Representative: Dorothy Martin
Bellevue High School, Bellevue, WA
to balance my checkbook. SIAM Representative: Terry Herdman
Virginia Tech Interdisplinary Center for Applied
Mathemathics, Blacksburg, VA
I think about checkbooks. Publications: Pat Bowler Johnson
New Trier High School, Winnetka, IL
Once in a while
Executive Assistant: Diane Rubin
I balance mine,
just like sometimes
I dust high shelves.
President: Alex Lee
--Joanne Growney
Joanne Growney is a former mathematics professor at
Bloomsburg University and is author of My Dance is
Mathematics, a collection of poems influenced by her
studies and experiences teaching math.
University of Oklahoma, Norman, OK
National Student Officers:
Vice-President:
Secretary-Treasurer:
Sergeant-at-Arms:
North Broward Preperatory, FL
StormLX@aol.com
Elaine Hart
Mount Rainer High School, WA
Megan Repar
Lafayette High School, LA
Nathaniel Watson
Farragut High School, TN

February 2002 The Mathematical Log Page 5
2001 Sister Scholastica Award Winner
Peggy Patout
Catholic High School
Baton Rouge, Louisiana
Mrs. Peggy Patout recently won the Sr.
Scholastica Award at the 2001 National Convention. As
the nominating student, I have the honor of writing an
article centered about this incredible person. It is my
intent to impart upon you in some small way the depth and
breadth of the impact that Mrs. Patout has had upon my
life and the lives of students in my high school. As a
member of Mu Alpha Theta for five years, I have spent
much of my high school career in the presence of Mrs.
Patout and have strong testimony as to her efforts in the
development of her students into complete men. It is my
intent to relate a few stories that will best convey Mrs.
Patout’s deep commitment to the mission of Mu Alpha
Theta and to the people within it.
In my eighth grade year at Catholic High School,
the Mu Alpha Theta organization under Mrs. Patout’s
guidance decided to host a math tournament for local
middle and junior-high school students. The motivation
for this tournament came purely from the desire to spread
interest in mathematics to students who have not yet
reached high school. Mrs. Patout has spent her years as
sponsor of Mu Alpha Theta attempting to imbue the
members of the CHS chapter with a sharp interest in
mathematics and in the teaching of mathematics. She
joined that message with one of service to the community
through tournaments like this. It is evident by the spirit of
her members that she has been tremendously successful in
this endeavor.
Mrs. Patout has the belief that all students can
become interested in math if it is taught in the correct
manner. Mrs. Patout felt so strongly about this that she
requested to not teach just honors students. Every year
Mrs. Patout teaches academic level students, honors
students, and AP level students. She achieves tremendous
success in each level. AP Scores and spring testing hours
for her AP and honors classes are her marks of success.
The most amazing indication of achievement has a much
greater impact on the world than the level of mastery she
teaches her students. In the past five years that I have
been associated with her, she has inspired numerous
students towards the pursuit of teaching degrees in
mathematics. Many more students she has simply taught
to admire teachers. Each student has no problem telling
you that Mrs. Patout is the reason for his career choice.
This is perhaps the greatest illustration of the dedication,
strength, and skill that Mrs. Patout possesses.
On a personal note, I have worked for several
years with her as an officer in Mu Alpha Theta. Because
of our work together, I can speak from experience when I
say that I have
seen no sponsor
anywhere who is
more dedicated,
kind, caring, and
influential than
Mrs. Patout. I owe
her a great debt
for the past five
years of my life.
Her desire is for
me to become a
productive and
caring member of
society. That is all
that she wants in
the way of repayment for her work. That, perhaps, says
the most about her character.
Perhaps the greatest thing that I take with me to
college from my years with Mrs. Patout is a simple
concept that has changed my whole way of thinking. She
has told me time and time again that awards should never
be the aim of your work. She believes that the true reward
of the things that she does lies in the knowledge that she
has changed the life of another person. With all of this
said, it is easy to see why Mrs. Patout is someone who
deserves with every ounce of meaning the Sister
Scholastica award, whether she would like to admit it or
not.
David Smith
Catholic High School
Baton Rouge, LA
Do you know a Mu Alpha Theta sponsor like Mrs. Patout Go to the Mu Alpha
Theta website and check under “Awards and Scholarships” to find out more
about how the history of the Sister Scholastica Award and how you can
nominate your sponsor for this award!
http://www.mualphatheta.org/awardships/awards.html

February 2002 The Mathematical Log Page 6
Math Students Race to Turn In…
Toilet Paper
by Seema Kakad
Myers Park High School
Charlotte, North Carolina
The Math Honor Society at Myers Park High
School has a new face. Suddenly, students aren't just
running to Mr. Aizenman's room to turn in late candy
money or to get their ticket for the next pizza party--
they're running to turn in… toilet paper That's right.
Mu Alpha Theta, traditionally a social club for
students who enjoy math, has turned into a social and
service club under the leadership of Advisor Ben
Aizenman, President Charlie Odom, Vice-President
Seema Kakad, Secretary Mary Werden, and
Treasurer Mousumi Medda.
For the November service project, over one
hundred math students in the 11th and 12th grade
brought in over 500 toiletry items for the Men's
Shelter in downtown Charlotte. Items ranged from
toothbrushes and deodorants to, yes, even toilet
paper. The officers delivered all six box loads to the
Men's Shelter on Tuesday, December 4th, to discover
that the shelter had just run out of toiletry items-- this
new delivery was perfect timing.
So what's next for the Math Honor Society
Students are already racing to Mr. Aizenman's room
in December to turn in canned goods for needy
families during the holiday time. For the new year
they plan to have a blanket drive and possibly
volunteer at a soup kitchen. The club also plans to
continue its popular dinners and has even rented out a
roller skating rink for the January social.
Who would have thought that students could
enjoy math, have fun, and give back to their
community The students of Mu Alpha Theta!
Of special note…
All members of Myers Park High School Mu Alpha
Theta Chapter must sign up to donate time to such
places as the Homeless Shelter, Christmas Bureau,
tutoring, and bring in toiletries, canned goods, and do
other service projects to retain membership. In
addition each year the club tries to give something
back to the school as a token of thanks. Money has
been donated for scholarships besides donating a
copy machine and calculators, for student use, to the
math department.
Benjamin Aizenman
Sponsor
On the Lighter Side…..
---Why are carrots round
Because they are not square roots.
---What did all the parallelograms take to the game
A rhombus!
--- What are side-by-side train tracks in Chicago
called
A pair of Els.
---Pythagoras: What’s your parabola
---Einstein: I didn’t sleep very well…
---Pythagoras: Oh, well I slept like a natural log.
Does your chapter have a
mascot
Above is the mascot of the Mu Alpha Theta Chapter
of Southside High School in Fort Smith, Arkansas. If
you have a mascot, please send a picture of it to the
National Office at nationaloffice@mualphatheta.org.
What would you like to see in the
Mathematical Log
Articles written by Mu Alpha Theta members or sponsors Math
applications Technology articles Interesting math concepts Puzzles
Articles about chapters Let me know what you’d like to see! Send your
comments and any topics you would like to see in the Mathematical Log
to:
Pat Bowler Johnson
New Trier High School
385 Winnetka Avenue
Winnetka, Il 60093
email: bowlerjp@newtrier.k12.il.us