A Mathematica Package for the Construction and Analysis of Kirkman Triple Systems, Journal of Statistical Software, Volume 11, Issue 7, 2004.
[3] Douglas B. West, Introduction to Graph Theory, Upper Saddle River, NJ: Prentice Hall, 1996.
Internet of things - IoT - Things are talking to InternetJyotindra Zaveri
Internet of things - IoT and Future store - inventory management - easy shopping without queue at the Point of Sale (POS). Consumer demand for convenience, product availability, and both personalized and contextualized interactions will drive retailers to adopt multiple IoT technologies in the coming years. Zaveri in this presentation is explaining IoT.
Shopping Mall with smart shelf can call for replenishment automatically - Smart Cart is interactive touch-screen with an ERP Server Computer, linked to the shopping cart handle and accesses a wireless in-store network. It makes shopping an easy, fun and enjoyable. Also retailers and brand marketers can advertise their products in an efficient way.
Real time inventory management - Data entry of items can be done automatically in the ERP system using an RFID Tag (Radio Frequency Identification).
I have uploaded related videos on YouTube. Click now https://goo.gl/FTMCwd
Kaip ženkliai padidinti pardavimus vidaus ir eksporto rinkose (II dalis)commonsenseLT
"Verslo žinių" kvietimu savo patirtimi dalinasi vienos sėkmingiausių eksporto bendrovių „Rifas” gen. direktorius Aidas Šetikas ir Mindaugas Voldemaras, vadovaujantis eksporto didinimo paslaugas teikiančiai įmonei „TOC Sales and Marketing”.
(įrašas darytas Kaune 2015 m. gruodžio 11 d.)
- Dvi pardavimų strategijos: kai turi laisvų gamybinių pajėgumų ir kai jų neturi;
- Kodėl verta tarti “sudie” senam geram klientui (arba kodėl miegama su atšalusiais lavonais);
- Matai, rodikliai ir sprendimai pardavimuose: ką žinoti vadovui, ir ko geriau nežinoti;
- Kodėl CRM įsidiegimas niekada nepadidina pardavimų ir kodėl magnetinė lenta kur kas geriau (demonstracija);
- Iš kur gauti potencialių klientų kontaktus ir ką daryti vėliau, kai jie nekelia ragelio ir neatsako į laiškus;
- Kiek kainuoja eksporto pardavimai vidutinio dydžio Lietuvos įmonei ir kiek trunka laukti „break even” (realūs pavyzdžiai);
- Penki „taip, bet”, stabdantys lietuviškos įmonės eksporto augimo tempus.
Dr. Roitman discusses the use of Artificial Intelligence to solve complex and insoluble problems. Artificial intelligence approach is in the root of I Know First predictive algorithm.
Internet of things - IoT - Things are talking to InternetJyotindra Zaveri
Internet of things - IoT and Future store - inventory management - easy shopping without queue at the Point of Sale (POS). Consumer demand for convenience, product availability, and both personalized and contextualized interactions will drive retailers to adopt multiple IoT technologies in the coming years. Zaveri in this presentation is explaining IoT.
Shopping Mall with smart shelf can call for replenishment automatically - Smart Cart is interactive touch-screen with an ERP Server Computer, linked to the shopping cart handle and accesses a wireless in-store network. It makes shopping an easy, fun and enjoyable. Also retailers and brand marketers can advertise their products in an efficient way.
Real time inventory management - Data entry of items can be done automatically in the ERP system using an RFID Tag (Radio Frequency Identification).
I have uploaded related videos on YouTube. Click now https://goo.gl/FTMCwd
Kaip ženkliai padidinti pardavimus vidaus ir eksporto rinkose (II dalis)commonsenseLT
"Verslo žinių" kvietimu savo patirtimi dalinasi vienos sėkmingiausių eksporto bendrovių „Rifas” gen. direktorius Aidas Šetikas ir Mindaugas Voldemaras, vadovaujantis eksporto didinimo paslaugas teikiančiai įmonei „TOC Sales and Marketing”.
(įrašas darytas Kaune 2015 m. gruodžio 11 d.)
- Dvi pardavimų strategijos: kai turi laisvų gamybinių pajėgumų ir kai jų neturi;
- Kodėl verta tarti “sudie” senam geram klientui (arba kodėl miegama su atšalusiais lavonais);
- Matai, rodikliai ir sprendimai pardavimuose: ką žinoti vadovui, ir ko geriau nežinoti;
- Kodėl CRM įsidiegimas niekada nepadidina pardavimų ir kodėl magnetinė lenta kur kas geriau (demonstracija);
- Iš kur gauti potencialių klientų kontaktus ir ką daryti vėliau, kai jie nekelia ragelio ir neatsako į laiškus;
- Kiek kainuoja eksporto pardavimai vidutinio dydžio Lietuvos įmonei ir kiek trunka laukti „break even” (realūs pavyzdžiai);
- Penki „taip, bet”, stabdantys lietuviškos įmonės eksporto augimo tempus.
Dr. Roitman discusses the use of Artificial Intelligence to solve complex and insoluble problems. Artificial intelligence approach is in the root of I Know First predictive algorithm.
Lab 05 – Gravitation and Keplers Laws Name __________________.docxDIPESH30
Lab 05 – Gravitation and Keplers Laws Name: _____________________
Why everyone in this class is attracted to everyone else.
https://phet.colorado.edu/en/simulation/gravity-force-lab
Adapted from Chris Bier’s Collisions PhET Lab OPTION A: CREATIVE COMMONS - ATTRIBUTION
Introduction:
Every object around you is attracted to you. In fact, every object in the galaxy is attracted to every other object in the galaxy. Newton postulated and Cavendish confirmed that all objects with mass are attracted to all other objects with mass by a force that is proportional to their masses and inversely proportional to the square of the distance between the objects' centers. This relationship became Newton's Law of Universal Gravitation. In this simulation, you will look at two massive objects and their gravitational force between them to observe G, the constant of universal gravity that Cavendish investigated.
Important Formulas:
Procedure: https://phet.colorado.edu/en/simulation/gravity-force-lab
1. Take some time and familiarize yourself with the simulation. Notice how forces change as mass changes and as distance changes.
2. Fill out the chart below for the two objects at various distances.
3. Rearranging the equation for Force, you can CALCULATE the value of G using the values given below for m1, m2, and d, and the value for the Force that you obtain in the simulation. Record the force between the two object and then solve (calculate G) for the universal gravitation constant, G and compare it to values published in books, online, or your text book. The numbers you calculate for G will vary slightly from row to row. Remember significant digits!15 pts
Mass Object 1 Mass Object 2 Distance Force Gravitation Constant,G
50.00 kg
25.00 kg
3.0m
50.00 kg
25.00 kg
4.0m
50.00 kg
25.00 kg
5.0m
50.00 kg
25.00 kg
6.0m
50.00 kg
25.00 kg
9.0m
What do you notice about the force that acts on each object? 3 pts
[Answer Here]
Average value of G: _________________2 ptsUnits of G: _______________2 pts
Published value of G: ________________2 pts Source: _______________2 pts
How did your average value of G compare to the published value for G that you found? 3 pts
[Answer Here]
Conclusion Questions and Calculations:Bold and Underlinethe correct answer to each question.
1. Gravitational force is always attractive / repulsive. (circle) 2 pts
2. Newton’s 3rd Law tells us that if a gravitational force exists between two objects, one very massive and one less massive, then the force on the less massive object will be greater than / equal to / less than the force on the more massive object. 2 pts
3. The distance between masses is measured from their edges between them / from their centers / from the edge of one to the center of the other. 2 pts
4. As the distance between masses decreases, force increases / decreases. 2 pts
5. Doubling the mass of both masses would result in a change of force between the mas ...
Time Table Scheduling Problem Using Fuzzy Algorithmic ApproachWaqas Tariq
Abstract In this paper we develop an algorithm to generate a course Time table using fuzzy algorithmic approach satisfying certain constraints. With an example we show that how these constraints are satisfied.
Lembar kerja peserta didik 1 materi spltv sma kelas xMartiwiFarisa
LKPD ini bertujuan untuk mengukur pengetahuan dan keterampilan peserda didik dalam menyelesaikan permasalahan yang berkaitan dengan sistem persamaan linear tiga vaeriabel.
Ph2A Win 2020 Numerical Analysis Lab
Max Yuen
Mar 2020
(use g = 9.8m/s2 for all problems.)
Background
Many physics problems cannot be solved directly by hand or analytically. We resort to numerical
methods to give us approximations to the problem. In this lab you will learn the Euler method,
which allows you to solve Newton’s laws of motion. This is done by treating the velocity as a
piecewise linear function with many time intervals and during interval the acceleration is assumed
to be uniform. This allows us to use the kinematic equations we learned about in the first half
of the class to approximate the motion. If we choose to partition the motion into smaller time
intervals, the approximation becomes much better since the differences between adjacent intervals
become smaller. In this lab, this numerical analysis method will be applied to the motion of a
falling object under the influence of gravity and drag force. If you are adventurous, you can even
try to extend this to 2D and compute the realistic trajectory of a baseball. You might even try
some other problems, such as a mass attached to a spring.
Euler’s Method Foundations
This method is well suited for problems where the acceleration is a function of the velocity, as in
the case of a falling object under the influence of gravity and drag:
a = f(v) (1)
Falling object with drag force
The model for drag fits the prescription for using Euler’s method since the net force on a falling
object with drag is given by:
ma = −mg −FD (2)
ma = −mg −
1
2
ρairACDv
2 · sgn(v) (3)
a = −g
(
1 +
ρairACDv
2 · sgn(v)
2mg
)
(4)
a = f(v) ← Equation of Motion (5)
where m is the mass of the falling object, a is the acceleration of the object (which is positive when
pointed up), ρair is the density of air (about 1.29 ·10−3kg/m3), A is the cross-sectional area, CD is
the drag coefficient, v is the object’s velocity, and sgn(v) is the signum function which returns the
sign of the argument. The second signum function is there to guarantee that the direction of the
drag force is always in the opposite direction of the velocity function. Note that we see that the
acceleration is an explicit function of v, which sort of makes this a chicken or egg problem. This is
because we need a to get v, but to get a we need v, so which one do we compute first? Hold that
thought. We’ll talk more on how to program this in EXCEL or Google Sheets later.
1
Figure 1: FBD for an object falling under the pull of gravity and resistance by drag force
Terminal Velocity
In lecture, we talked about how after waiting for some time, if the object started at rest the
speed will increase and the drag force will also become larger and eventually balance out with the
gravitational force. When this happens, we have reached terminal velocity vterm = −v. This can
be solved by setting a = 0:
0 = −mg −
1
2
ρairACDv
2 · sgn(v) (6)
2mg = ρairACDv
2
term (7)
→ vterm =
√
2mg
ρairACD
(8)
Using this definition for the terminal ...
1. Assume that an algorithm to solve a problem takes f(n) microse.docxSONU61709
1. Assume that an algorithm to solve a problem takes f(n) microseconds for some function f of the input size n. For each time t labeled across the top, determine the exact largest value of n which can be solved in time f(n) where f(n) ≤ t. Use a calculator! You will find it helpful to convert the t values to microseconds, and you may find it helpful to insert a row for n. Note that “lg n” is the log2 n. Note that the only row you can’t write out the values for fully is the “lg n” row—only there may you write 2x for the appropriate value of x. Use the Windows built-in scientific calculator (under Accessories menu) as necessary. A couple values are filled in to get you started. Important: “exact values” means precisely that. Check your answers with values above and below!
Time t =
f(n) =
1 second
1 hour
1 day
1 month
=30 days
n2
1,609,968
lg n
n3
2n
n lg n
2,755,147,513
2. Use loop counting to give a O( ) characterization of each of the following loops basing each upon the size of its input:
a. Algorithm Loop1(n):
s ← 0
for i ← 1 to n do
s ← s + i
b. Algorithm Loop2(p):
p ← 1
for i ← 1 to 2n do
p ← p * i
c. Algorithm Loop3(n):
p ← 1
for j ← 1 to n2 do
p ← p * i
d. Algorithm Loop4(n):
s ← 0
for j ← 1 to 2n do
for k ← 1 to j do
s ← s + j
e. Algorithm Loop5(n):
k ← 0
for r ← 1 to n2 do
for s ← 1 to r do
k ← k + r
3. Order the following functions from smallest to largest by their big-O notation—you can use the letters in your answer rather than copying each formula. Be clear which is smallest and which is largest, and which functions are asymptotically equivalent. For example, if g, h, and m are all O(n lg n), you would write g = h = m = O(n lg n).
a. 562 log3 108
b. n3
c. 2n lg n
d. lg nn
e. n3 lg n
f. (n3 lg n3)/2
g. nn
h. 56n
i. log5 (n!)
j. ncos n
k. n / lg n
l. lg* n
m.
4. a. Which of these equations is true, and why?
b. Which of these is smaller for very large n?
Trisecting the Circle: A Case for Euclidean Geometry
Author(s): Alfred S. Posamentier
Source: The Mathematics Teacher, Vol. 99, No. 6 (FEBRUARY 2006), pp. 414-418
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/27972006
Accessed: 09-02-2018 18:19 UTC
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide
range of content in a trusted digital archive. We use information technology and tools to increase productivity and
facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
http://about.jstor.org/terms
National Council of Teachers of Mathematics is collaborating with JSTOR to digitize,
preserve and extend access to The Mathematics Teacher
This content downloaded ...
Lab 05 – Gravitation and Keplers Laws Name __________________.docxDIPESH30
Lab 05 – Gravitation and Keplers Laws Name: _____________________
Why everyone in this class is attracted to everyone else.
https://phet.colorado.edu/en/simulation/gravity-force-lab
Adapted from Chris Bier’s Collisions PhET Lab OPTION A: CREATIVE COMMONS - ATTRIBUTION
Introduction:
Every object around you is attracted to you. In fact, every object in the galaxy is attracted to every other object in the galaxy. Newton postulated and Cavendish confirmed that all objects with mass are attracted to all other objects with mass by a force that is proportional to their masses and inversely proportional to the square of the distance between the objects' centers. This relationship became Newton's Law of Universal Gravitation. In this simulation, you will look at two massive objects and their gravitational force between them to observe G, the constant of universal gravity that Cavendish investigated.
Important Formulas:
Procedure: https://phet.colorado.edu/en/simulation/gravity-force-lab
1. Take some time and familiarize yourself with the simulation. Notice how forces change as mass changes and as distance changes.
2. Fill out the chart below for the two objects at various distances.
3. Rearranging the equation for Force, you can CALCULATE the value of G using the values given below for m1, m2, and d, and the value for the Force that you obtain in the simulation. Record the force between the two object and then solve (calculate G) for the universal gravitation constant, G and compare it to values published in books, online, or your text book. The numbers you calculate for G will vary slightly from row to row. Remember significant digits!15 pts
Mass Object 1 Mass Object 2 Distance Force Gravitation Constant,G
50.00 kg
25.00 kg
3.0m
50.00 kg
25.00 kg
4.0m
50.00 kg
25.00 kg
5.0m
50.00 kg
25.00 kg
6.0m
50.00 kg
25.00 kg
9.0m
What do you notice about the force that acts on each object? 3 pts
[Answer Here]
Average value of G: _________________2 ptsUnits of G: _______________2 pts
Published value of G: ________________2 pts Source: _______________2 pts
How did your average value of G compare to the published value for G that you found? 3 pts
[Answer Here]
Conclusion Questions and Calculations:Bold and Underlinethe correct answer to each question.
1. Gravitational force is always attractive / repulsive. (circle) 2 pts
2. Newton’s 3rd Law tells us that if a gravitational force exists between two objects, one very massive and one less massive, then the force on the less massive object will be greater than / equal to / less than the force on the more massive object. 2 pts
3. The distance between masses is measured from their edges between them / from their centers / from the edge of one to the center of the other. 2 pts
4. As the distance between masses decreases, force increases / decreases. 2 pts
5. Doubling the mass of both masses would result in a change of force between the mas ...
Time Table Scheduling Problem Using Fuzzy Algorithmic ApproachWaqas Tariq
Abstract In this paper we develop an algorithm to generate a course Time table using fuzzy algorithmic approach satisfying certain constraints. With an example we show that how these constraints are satisfied.
Lembar kerja peserta didik 1 materi spltv sma kelas xMartiwiFarisa
LKPD ini bertujuan untuk mengukur pengetahuan dan keterampilan peserda didik dalam menyelesaikan permasalahan yang berkaitan dengan sistem persamaan linear tiga vaeriabel.
Ph2A Win 2020 Numerical Analysis Lab
Max Yuen
Mar 2020
(use g = 9.8m/s2 for all problems.)
Background
Many physics problems cannot be solved directly by hand or analytically. We resort to numerical
methods to give us approximations to the problem. In this lab you will learn the Euler method,
which allows you to solve Newton’s laws of motion. This is done by treating the velocity as a
piecewise linear function with many time intervals and during interval the acceleration is assumed
to be uniform. This allows us to use the kinematic equations we learned about in the first half
of the class to approximate the motion. If we choose to partition the motion into smaller time
intervals, the approximation becomes much better since the differences between adjacent intervals
become smaller. In this lab, this numerical analysis method will be applied to the motion of a
falling object under the influence of gravity and drag force. If you are adventurous, you can even
try to extend this to 2D and compute the realistic trajectory of a baseball. You might even try
some other problems, such as a mass attached to a spring.
Euler’s Method Foundations
This method is well suited for problems where the acceleration is a function of the velocity, as in
the case of a falling object under the influence of gravity and drag:
a = f(v) (1)
Falling object with drag force
The model for drag fits the prescription for using Euler’s method since the net force on a falling
object with drag is given by:
ma = −mg −FD (2)
ma = −mg −
1
2
ρairACDv
2 · sgn(v) (3)
a = −g
(
1 +
ρairACDv
2 · sgn(v)
2mg
)
(4)
a = f(v) ← Equation of Motion (5)
where m is the mass of the falling object, a is the acceleration of the object (which is positive when
pointed up), ρair is the density of air (about 1.29 ·10−3kg/m3), A is the cross-sectional area, CD is
the drag coefficient, v is the object’s velocity, and sgn(v) is the signum function which returns the
sign of the argument. The second signum function is there to guarantee that the direction of the
drag force is always in the opposite direction of the velocity function. Note that we see that the
acceleration is an explicit function of v, which sort of makes this a chicken or egg problem. This is
because we need a to get v, but to get a we need v, so which one do we compute first? Hold that
thought. We’ll talk more on how to program this in EXCEL or Google Sheets later.
1
Figure 1: FBD for an object falling under the pull of gravity and resistance by drag force
Terminal Velocity
In lecture, we talked about how after waiting for some time, if the object started at rest the
speed will increase and the drag force will also become larger and eventually balance out with the
gravitational force. When this happens, we have reached terminal velocity vterm = −v. This can
be solved by setting a = 0:
0 = −mg −
1
2
ρairACDv
2 · sgn(v) (6)
2mg = ρairACDv
2
term (7)
→ vterm =
√
2mg
ρairACD
(8)
Using this definition for the terminal ...
1. Assume that an algorithm to solve a problem takes f(n) microse.docxSONU61709
1. Assume that an algorithm to solve a problem takes f(n) microseconds for some function f of the input size n. For each time t labeled across the top, determine the exact largest value of n which can be solved in time f(n) where f(n) ≤ t. Use a calculator! You will find it helpful to convert the t values to microseconds, and you may find it helpful to insert a row for n. Note that “lg n” is the log2 n. Note that the only row you can’t write out the values for fully is the “lg n” row—only there may you write 2x for the appropriate value of x. Use the Windows built-in scientific calculator (under Accessories menu) as necessary. A couple values are filled in to get you started. Important: “exact values” means precisely that. Check your answers with values above and below!
Time t =
f(n) =
1 second
1 hour
1 day
1 month
=30 days
n2
1,609,968
lg n
n3
2n
n lg n
2,755,147,513
2. Use loop counting to give a O( ) characterization of each of the following loops basing each upon the size of its input:
a. Algorithm Loop1(n):
s ← 0
for i ← 1 to n do
s ← s + i
b. Algorithm Loop2(p):
p ← 1
for i ← 1 to 2n do
p ← p * i
c. Algorithm Loop3(n):
p ← 1
for j ← 1 to n2 do
p ← p * i
d. Algorithm Loop4(n):
s ← 0
for j ← 1 to 2n do
for k ← 1 to j do
s ← s + j
e. Algorithm Loop5(n):
k ← 0
for r ← 1 to n2 do
for s ← 1 to r do
k ← k + r
3. Order the following functions from smallest to largest by their big-O notation—you can use the letters in your answer rather than copying each formula. Be clear which is smallest and which is largest, and which functions are asymptotically equivalent. For example, if g, h, and m are all O(n lg n), you would write g = h = m = O(n lg n).
a. 562 log3 108
b. n3
c. 2n lg n
d. lg nn
e. n3 lg n
f. (n3 lg n3)/2
g. nn
h. 56n
i. log5 (n!)
j. ncos n
k. n / lg n
l. lg* n
m.
4. a. Which of these equations is true, and why?
b. Which of these is smaller for very large n?
Trisecting the Circle: A Case for Euclidean Geometry
Author(s): Alfred S. Posamentier
Source: The Mathematics Teacher, Vol. 99, No. 6 (FEBRUARY 2006), pp. 414-418
Published by: National Council of Teachers of Mathematics
Stable URL: http://www.jstor.org/stable/27972006
Accessed: 09-02-2018 18:19 UTC
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide
range of content in a trusted digital archive. We use information technology and tools to increase productivity and
facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
http://about.jstor.org/terms
National Council of Teachers of Mathematics is collaborating with JSTOR to digitize,
preserve and extend access to The Mathematics Teacher
This content downloaded ...
1. Finding Lab Groups using Combinatorial
Designs
Eddie Cardwell
Christopher Newport University
April 15, 2015
1
2. Combinatorial design theory is a field of mathematics that relates to
questions about wether it is possible to arrange elements of a finite set into
subsets so that certain properties hold. Many types of these designs that
we study today were first seen as mathematical puzzles or brain-teasers that
were posed in the eighteenth and nineteenth century. However, the actual
study of design theory as a mathematical discipline began in the twentieth
century due to its applications of lotteries, algorithms, networking, cryptog-
raphy, and scheduling.[3]
Steiner Triple Systems were first defined by W.S.B. Woolhouse in 1844
as a prize question in the Ladys and Gentlemans Diary. Thomas Kirkman
solved the prize question in 1847. Three years later, Kirkman posed a varia-
tion of Woolhouse s question that came to be known as Kirkman s schoolgirl
problem.[1] He asked: Is it possible for a schoolmistress to take 15 girls on
a walk each day of the week, walking with five rows of three girls each, in
such a way that each pair of girls walks together in the same row on exactly
one day? Kirkman solved his own problem in 1850 by using a variation of a
Steiner triple System, which became known as a Kirkman Triple System.
A Steiner Triple System is a type of block design named after Jakob
Steiner. It is an ordered pair (S, T), where S is a finite set of points, and T
is a set of 3 element subsets called triples, such that each pair of distinct
elements of S occurs in exactly one triple of T. The order of a Steiner Triple
System is equivalent to the size of S. Steiner Triple Systems are often de-
noted STS(n) for a Steiner Triple System of order n.
2
3. Definition 1: A parallel class in a STS(S, T) is a set of triples in T that
partitions the set.
Definition 2: A STS(S, T) is resolvable if the triples in T can be partitioned
into parallel classes.
A Kirkman Triple System, denoted KTS(n), is a resolvable Steiner Triple
system with an added property of being resolvable. In a KTS(n), the num-
ber of triples in each parallel class is n
3
because 3 students are in each triple.
The number of parallel classes is n 1
2
because each student works with 2 other
students at a time, and needs to work with all n 1 others.
Noting that for all v greater than or equal to 1, there exists a KTS(6v+3);
it is easily deduced that the smallest non-trivial example of a Kirkman triple
system is KTS(9) which is shown below.
3
4. Example 1: A KTS(9) with parallel classes ⇡1 ⇡2 ⇡3 ⇡4:
1 2 3
4 6
7 8 9
5
1 2 3
4 6
7 8 9
5
1 2 3
4 6
7 8 9
5
1 2 3
4 6
7 8 9
5
⇡1
1 2 3
4 5 6
7 8 9
⇡2
1 4 7
2 5 8
3 6 9
⇡3
1 6 8
2 4 9
3 5 7
⇡4
1 5 9
2 6 7
3 4 8
The solution to the Kirkman schoolgirl problem is indeed a KTS(15).
The parallel classes in the KTS(15) can be thought of as the arrangement
of the girls on any day of the week. Kirkmans publication of the 15-girl so-
lution was just the beginning of what we now know as combinatorial design
theory.[2]
We initially examined a class of 30 students that would be divided into
groups of 3 for each week of a semester. Every week the groups of 3 would
change and we wanted to see how many weeks the class could go with the
condition that no student could work with the same person more than once.
The di culty for this particular problem arises from the class size. Each
student must work with 29 others, while seeing each of them two at a time.
4
5. This observation gives an upper bound of 14 weeks with non repeating pairs,
getting a full semester out of it is challenging. Since we were working with
30 students, which is not of the form 6n + 3 there is no complete KTS(30).
We began our solution by letting each student correspond to a number.
In this case since we were working with 30 students, we labeled the stu-
dents 0 to 29. Our first attempt was to see how many weeks we could get
with a method of simply o↵setting the number in between each student. For
instance, we started with an o↵set of 1. This o↵set gave us a first week of
students (1,2,3), (4,5,6), (7,8,9), (10,11,12), (13,14,15), (16,17,18), (19,20,21),
(22,23,24), (25,26,27), and (28,29,0).
Then, setting that as our Week 1 we changed the o↵set from 1 to other
numbers with the condition that the o↵set would be mod 30, and negative
o↵sets would be shifts of the other direction. We discovered that this method
was ine cient and we inevitably ran into conflicts starting around week 6.
Given that we were running into repetition conflicts so early on in our con-
struction, we decided to modify the constraints on our problem.
We modified our original problem to the see how many weeks were possi-
ble to go with groups of 3 students, but with the additional constraint that
no student could work with the same student more than twice rather than
once. We started with a shift of 1, and continued with di↵erent o↵sets until
we ran into a repeat. We found that with o↵sets of 1, 2, 5, 7, 8, 10, 11, 13,
14, 17, 19, 23, 25, 26, and 29, we are able to get 15 weeks. Looking through
these weeks, we noticed hat there are some groups that never work together.
5
6. Set 1 Set 2 Set 3
0 1 2
3 4 5
6 7 8
9 10 11
12 13 14
15 16 17
18 19 20
21 22 23
27 28 20
Then, week 1 is taken out in order to use its triples along with the stu-
dents above that have not worked together. Various shifts were used on
the sets that have never worked together to try to get more weeks with out
letting a student work with someone more than twice. When there was a
collision, we changed the starting points of the cycle such that the colli-
sion was avoided. For sets {0, 1, 2}, {3, 4, 5}, {6, 7, 8} we used an o↵set of 3.
For {9, 10, 11), {18, 19, 20}, {27, 28, 29} we used an o↵set of 9. This method,
along with the first method yields 20 weeks of the following construction:
6
8. {18, 13, 8}, {4, 29, 24}, {19, 14, 9}, {4, 29, 24}, {19, 14, 9}
Week 13:{1, 27, 23}, {19, 15, 11}, {7, 3, 29}, {25, 21, 17}, {13, 9, 5},
{2, 28, 24}, {20, 16, 12}, {8, 4, 0}, {26, 22, 18}, {14, 10, 6}
Week 14:{1, 0, 29}, {28, 27, 26}, {25, 24, 23}, {22, 21, 20}, {19, 18, 17},
{16, 15, 14}, {13, 12, 11}, {10, 9, 8}, {7, 6, 5}, {4, 3, 2}
Week 15:{0, 1, 2}, {3, 6, 9}, {4, 7, 10}, {5, 8, 11}, {12, 15, 18},
{13, 16, 19}, {14, 17, 20}, {21, 24, 27}, {22, 25, 28}, {23, 26, 29}
Week 16:{3, 4, 5}, {6, 9, 12}, {7, 10, 13}, {8, 11, 14}, {15, 18, 21},
{16, 19, 22}, {17, 20, 23}, {24, 27, 0}, {25, 28, 1}, {26, 29, 2}
Week 17:{6, 7, 8}, {9, 12, 15}, {10, 13, 16}, {11, 14, 17}, {18, 21, 24},
{19, 22, 25}, {20, 23, 26}, {27, 0, 3}, {28, 1, 4}, {29, 2, 5}
Week 18:{9, 10, 11}, {12, 21, 0}, {13, 22, 1}, {14, 23, 2}, {15, 24, 3},
{16, 25, 4}, {17, 26, 5}, {18, 27, 6}, {19, 28, 7}, {20, 29, 8}
Week 19:{18, 19, 20}, {21, 0, 9}, {22, 1, 10}, {23, 2, 11}, {24, 3, 12},
{25, 4, 13}, {26, 5, 14}, {27, 6, 15}, {28, 7, 16}, {29, 8, 17}
Week 20:{27, 28, 29}, {30, 9, 18}, {1, 10, 19}, {2, 11, 20}, {3, 12, 21},
{4, 13, 22}, {5, 14, 23}, {6, 15, 24}, {7, 16, 25}, {8, 17, 26}
By adding the constraint that a student may work with another stu-
dent no more than two times, there are opportunities for future work. In
our construction, we removed the first week in order to gain 6 more weeks
by finding the sets of students that do not work together. It may be possible
to gain additional weeks by removing other weeks instead of week 1. Addi-
8
9. tionally, other popular lab sizes can have 24 or even 18 students. By using
our construction where pairs might work together twice it may be possible
to get a full semester out of a 24 or 18 student class.
9
10. References
[1] Roger Lindner, Design Theory, Boca Raton, Florida, CRC Press, 1997.
[2] Ezra Brown and Keith E. Mellinger, Kirkman’s Schoolgirls Wearing Hats
and Walking through Fields of Numbers, Mathematics Magazine Vol. 82,
2010.
[3] Douglas Stinson, Combinatorial Designs, New York, New York, Springer,
2003.
10
![Combinatorial design theory is a field of mathematics that relates to
questions about wether it is possible to arrange elements of a finite set into
subsets so that certain properties hold. Many types of these designs that
we study today were first seen as mathematical puzzles or brain-teasers that
were posed in the eighteenth and nineteenth century. However, the actual
study of design theory as a mathematical discipline began in the twentieth
century due to its applications of lotteries, algorithms, networking, cryptog-
raphy, and scheduling.[3]
Steiner Triple Systems were first defined by W.S.B. Woolhouse in 1844
as a prize question in the Ladys and Gentlemans Diary. Thomas Kirkman
solved the prize question in 1847. Three years later, Kirkman posed a varia-
tion of Woolhouse s question that came to be known as Kirkman s schoolgirl
problem.[1] He asked: Is it possible for a schoolmistress to take 15 girls on
a walk each day of the week, walking with five rows of three girls each, in
such a way that each pair of girls walks together in the same row on exactly
one day? Kirkman solved his own problem in 1850 by using a variation of a
Steiner triple System, which became known as a Kirkman Triple System.
A Steiner Triple System is a type of block design named after Jakob
Steiner. It is an ordered pair (S, T), where S is a finite set of points, and T
is a set of 3 element subsets called triples, such that each pair of distinct
elements of S occurs in exactly one triple of T. The order of a Steiner Triple
System is equivalent to the size of S. Steiner Triple Systems are often de-
noted STS(n) for a Steiner Triple System of order n.
2](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)