History of Mathematics Final Papers Juniata College Fall 2012 Dr. John Bukowski

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1 History of Mathematics Final Papers Juniata College Fall 2012 Dr. John Bukowski Contents: 1. Mayan Mathematics Nick Donlan 2. Pascal s Triangle Kevin Snyder 3. The Origins of Probability Theory! Amy Ankney 4. James Stirlin!g an!d N!umbers of the First an!d Secon!d Kin!d Matt Johann 5. Seven Bridges of Königsberg: An Eulerian Trail Cody Johnson 6. Lloyd Shapley and His Work on Game Theory Steve Kidhardt 7. The Four Color Problem Melissa Stanton

2 Nick Donlan History of Math Dr. Bukowski 12/13/12 Mayan Mathematics When people think about pre-columbian American cultures, three civilizations tend to come to mind. The first of these civilizations is the Aztecs who dominated most of central-south America from the 14 th to 16 th centuries. Another is the Incas, who were the largest of the three civilizations and came from the highlands of Peru in the early 13 th century. However, the third, and oldest, of the three was the Mayans. The Mayans were first established around 2000 BC, but they did not reach their pinnacle until around AD. The Mayans are known for many things including their art, architecture, astronomical developments, and calendar. They are also the only pre-columbian American civilization that developed a fully written language. However, it was their revolutionary mathematics that led to the Mayans most famous developments. In 1505, Hernan Cortes sailed from Spain and landed in Hispaniola, which is modern day Santo Domingo (the capital and largest city in the Dominican Republic). Cortes had heard the stories of Columbus s voyages to the New World and became very intrigued by the prospect of what lay in these new lands. Wanting to see these lands for himself, Cortes sailed to and lived in Hispaniola for a few years. Eventually he sailed to Cuba in 1511 with aspirations to conquer the land and its people. He was ultimately successful and elected leader of Santiago twice. However, he soon had aspirations for bigger and better things. So, on February 18, 1519, with an army of 11 ships, 508 soldiers, 100 sailors, and 16 horses, Cortes made his way to the coast of the

3 Yucatan peninsula, which is modern day Southeastern Mexico. He eventually landed on the Northern coast, at the city of Tabasco. Cortes was ready for a fight from the natives, but he surprisingly met very little resistance. In fact, not only did he meet virtually no resistance, they all but welcomed Cortes with open arms. They showered him with many gifts, and he met his eventual wife, Malinche, here as well. These people that Cortes had landed upon and conquered in Tabasco were the descendants of the ancient Mayans. When studying Mayan culture and more specifically their mathematics, it is important to note how knowledge of the Mayans and their achievements were made know to the rest of the world. Diego de Landa was a Spanish monk who belonged to the Franciscan order. When he was seventeen, he asked to be sent to the New World as a missionary. After gaining permission, he landed in the Yucatan Peninsula, where these Mayan descendants lived. He did his best to protect the indigenous people from new Spanish rulers, such as Cortes. He visited many of the Mayan cities and tried to learn as much about the people and their culture as he could, during his time with them. However, the one aspect of the people s culture that Landa absolutely abhorred was their religious practices. Landa was obviously a very devout Christian, and to Landa, the Mayan religion was the devil s work (Ifrah). Their religion consisted of a series of hieroglyphics and icons. Because of this, Landa ordered that all idols, religious works, and anything else related to the Mayan religion, be burned. However, for some reason, Landa then wrote and published a book in 1566, Relacion de las cosas de Yucatan, which gave detailed accounts of everything Mayan, including religion. It described their hieroglyphics, customs, temples, religious practices, and the overall history of the Mayans. Landa s book is a major reason why the world knows so much about the Mayan civilization today.

4 However, despite Landa s desperate attempts to eradicate many of the records of the Mayans, a small number of original Mayan documents survived. These also help to educate people about Mayan culture, society, and customs. The three most famous are the Paris Codex, the Madrid Codex, and the Dresden Codex. The Paris Codex is housed in the Bibliotheque nationale in Paris. The Madrid Codex is housed in the American Museum in Madrid, and the Dresden Codex, which is a piece on astronomy, is stored in the Sachsische Landesbibliothek Dresden. The ancient Mayans had a very sophisticated civilization for the time. They built very large cities, equipped with such fixtures as temples, palaces, shrines, plazas, and giant basins to hold rainwater. Astronomer priests ruled Mayan people using religion. Their farming system was very sophisticated, with raised fields and intricate irrigation systems. They were used to provide food for the massive amounts of people that lived in the cities. A common culture, calendar, and religious practice held the mighty and vast Mayan civilization together, but in order to develop a calendar and the astronomy behind their religious practices, a very good grasp of mathematics was required. The Mayans created a sophisticated number system more advanced than that of any other civilization in the world. Today, modern society for the most part uses a base ten system. There are also some uses of a sexagesimal system, or a base sixty number system, in modern society as well. This stems from the ancient Sumerians and Babylonians as they also used a base sixty system. There are sixty seconds in a minute, sixty minutes in an hour, etc. Virtually no one uses a base twenty or vigesimal system. However, this is precisely what the Mayans used. Although the reason why they used twenty as their base cannot be proved, it is thought, with fairly decent certainty, that it was because people counted using their fingers and toes. However, despite being a base twenty

5 system, the number five also played a major role in their number system. This can be attributed to the number of fingers or toes on each hand or foot. Somewhat surprisingly though, the Mayans only used three different symbols to represent numbers. They used a system of dots, bars, and a symbol resembling a seashell. However, there were two distinct features of the Mayans mathematics that set it apart from anything else in the world during this time. First, they discovered the idea of zero (which they denoted with their shell symbol) and second, their number system was positional in nature. The following is a visual example, from the St. Andrews history of mathematics website, of the Mayan positional number system:

6 However, despite this system being positional in nature, it is not a true positional number system. In a true base twenty system, the first number of the system would signify the number of units up to nineteen, the next would represent the number of 20 s up to nineteen, and then the next number would denote the number of 20 2 or 400 s up to nineteen. The Mayan system has a slight variation. Their system starts the same with the first two places, but the third placeholder in the Mayan system represents 360 instead of the 400 it would represent in a true base twenty system. After the third place, the system continues just like any other base twenty number system. So the fourth place denotes the number of 20 4 or 160,000 up to nineteen, the fifth place would denote the number 20 5 or 3,200,000 up to nineteen, etc. For example [5;12;6;3;14] represents This system was described in the Dresden Codex, and consequently the only system for which written evidence exists. The irregularities in this system are attributed to needing to use this number system for astronomical and calendar calculations, which do not fall perfectly into a base twenty system. However, it was thought a second base twenty system, which was an actual true vigesimal system, was used by merchants. This system also utilized special symbols for 20, 400, and Georges Ifrah, in his book A Universal History of Numbers: From Prehistory to the Invention of the Computer, wrote Even though no trace of it remains, we can reasonably assume that the Maya had a number system of this kind, and that intermediate numbers were figured by repeating the signs as many times as was needed. As previously mentioned, one of the main reasons the Mayans developed such a number system was for the development of their calendar. There were influences from other directions and sources, but the base twenty systems of the Mayans played a major role in the structure of their calendar.

7 The Mayans actually had two main calendars, both of which they observed. The shorter of the two is known as Tzolkin and consisted of 260 days. It was split into thirteen months with twenty days in each month. The thirteen months were named after the thirteen gods of the Mayan religion, and the twenty days were numbered from zero to nineteen. This makes the Mayan number system s influence on their calendar clear. Their second calendar consisted of 365 days and was called the Haab. This calendar consisted of eighteen months and was more focused around agricultural and religious events. Each month again consisted of twenty days numbered from zero to nineteen. However, doing the math, eighteen multiplied by twenty is 360. Therefore, there are five extra days left in the Haab calendar year. These extra five days made up a short month called the Wayeb. The Mayans hated and feared the Wayeb. They considered it extremely unlucky and did not wash, comb their hair, or do any work during these five days. It was also believed that any child born during the Wayeb would have bad luck, remain poor, and generally be very unhappy during their entire life. Why the Mayans had a calendar based on 260 days is not entirely known. One theory is the Mayans lived in a place that the sun was directly overhead every 260 days, with 105 days in between periods. Another theory is that the Mayans had thirteen gods, and twenty was a man s number, so by giving each Mayan god a twenty-day month, it gave a ritual calendar consisting of 260 days. Despite the reason for the two calendars, having them meant that they would coincide with each other after 18,980 days (which equates to fifty-two years on the Haab, or seventy-three years on Tzolkin). Another major aspect that contributed to the development of the calendar was the synodic period of Venus. The Mayans noticed that Venus returned to the same position every 584 days. Therefore, after only two of the of the fifty-two year cycles, Venus would have made

8 sixty-five revolutions and end up at the original position. This extraordinary occurrence would happen every 104 years and was marked by great celebrations. There was also a third method of measuring time that the Mayan people used. Although it wasn t a calendar, it still utilized their extensive mathematics. It was an absolute timescale, which began on a date from which days and times were measured going forward. The date most often thought of as the start day, although not unanimous among scholars, is August 12, 3113 BC. This method of counting days is known as the Long Count. The unique aspect of the Long Count was that it was based on neither the Tzolkin nor the Haab. Instead, it is based on a year of 360 days. This shows the most likely reason for the departure of the number system from the true vigesimal system; it was so that the system approximately represented years. Many examples of the Long Count were found in Mayan cities and towns, such as the date a certain building was completed. For example, a plate that came from the town Tikal reads [8;14;3;1;12]. Doing the math utilizing Mayan mathematics, this equates to, This is how many days away from the original date of August 12, 3113 BC that the plate was carved, meaning it was finished in 320 AD. However, despite the innovative methods of the Mayans, their number system did have some limitations. They did not seem to have any concept of the fraction, they could not divide or multiply, and because their number system was not a true base twenty system it lacked some of the nice properties that true base twenty systems have. For example, [9;8;9;13;0] equates to However, [9;8;9;13] is equal to

9 In a true base twenty system, moving all the numbers to the left would just multiply the final product by twenty, but. Finally, the Mayans used their number system and mathematics to make advances in astronomy. The Mayans carried out very accurate and astonishing observations with nothing other than sticks. They put these sticks in the form of crosses and used the right angle formed by them as viewing instruments to make calculations. With these crude and basic instruments, the Mayans were able to calculate the length of the year to be days. The modern length, calculated with much better technology, is days. The fact that the Mayans could get that accurate of an estimation using only sticks shows the true genius and power of Mayan mathematics. They also calculated the length of the lunar month with eerily similar precision. At Copan, Mayan astronomers found that 149 lunar months lasted 4400 days, which equates to days in a lunar month. At Palenque, a town in Tabasco, Mayan astronomers calculated that 81 lunar months lasted 2392 days, giving days as the length of the lunar month. The accepted modern day calculation gives days as the length of the lunar month. Clearly, these two calculations were extremely impressive achievements, especially with the limited resources and technology the Mayans had at their disposal. The Mayans were a powerful civilization in the pre-columbian Americas. However, unlike the other two powers of the time, the Incas and the Aztecs, the Mayan people never disappeared. Even with the arrival of Spanish conquistadors, who were the downfall of other civilizations such as the Aztecs, the Mayans continued to survive. Even today, there are still large groups of Mayans scattered throughout Central and South America. Mayan influence can still be felt in these areas, where millions still speak Mayan languages. However, it is their

10 contributions to mathematics that really made the Mayans stand out in the world and led to many of the developments and achievements that they are still known for today.

11 Bibliography G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998). "Mayan Mathematics." Mayan Mathematics. N.p., n.d. and.ac.uk/histtopics/mayan_mathematics.html. 12 Dec "Mayan Mathematics - The Story of Mathematics." Mayan Mathematics - The Story of Mathematics. N.p., n.d Dec

12 PASCAL S TRIANGLE KEVIN SNYDER History: Blaise Pascal was born June 19th, 1623 in Clermont, France, and died August 19th, 1662 in Paris, France[1]. He was only three years old when his mother passed away. His father, Etienne Pascal, had to raise him along with three other children after that [1]. He was the only son. His father was a local judge in Clermont [5] but later moved his family to Paris because of work [1]. His father was a big influence on his son s education. He did not take the typical approach. He decided it was best to educate Blaise himself. His father did not feel his son was going to be ready to learn math, however, so he did not have any mathematical texts in the house. He wanted to hold off on math until Blaise was 15 years old[1]. His father felt this way his son would not be stressed out over too much learning [5]. This did not stop Pascal from being curious in the subject. At age 12, he was very intrigued and curious about geometry [1]. He wanted to know more about the subject so he asked his father. Once Pascal learned about the subject he wanted to learn so much more about the subject. He then, on his own will, decided to give up his leisure time to study math [5]. Within weeks of this, Pascal came up with a proof that the sum of the angles in a triangle equaled the sum of two right angles [5]. From here, his father knew that he needed to teach his son more about math. His father did a few things to help his son further his education in mathematics. He let his son come to the Mersenne s meetings Date: December 13,

13 2 KEVIN SNYDER which were gatherings of a lot of mathematicians in Paris [5]. His father also gave his son a copy of Euclid s Elements which was a big inspiration for Pascal[1]. In 1639, the Pascal family was forced to move again. His father got a new job in Rouen, France, where his father had been appointed as a tax collector [1]. Pascal wanted to help his father with his work so he actually created one of the first mechanical calculators. This calculator was specific to currency in France but was a big help to his father [1]. While in Rouen, Pascal actually was starting to focus his studies on analytical geometry and physics. He published one of his first works on atmospheric pressure [1]. He believed that there was a vacuum that existed above the atmosphere [1]. This idea was not accepted by many intellectuals around around the country. One in particular was Descartes. He visited Pascal once and they argued for two days straight about whether there is actually a vacuum[1]. He wrote a few other papers containing information about the vacuum afterwards. His father passed away in This day had a big effect on Pascal. After the death of his father, he wrote to his siblings on the meaning of death and started to become very religious and would change how he lived the rest of his life[1]. In 1654, Pascal wrote the Treatise on the Arithmetical Triangle. He was not the first to discuss the topic however his work is what made the triangle as popular as it is today [1]. His work led to Newton s discovery of the general binomial theorem for fractional and negative powers [1]. I will go into further detail on this document later. After this, Pascal started to work with Fermat on the foundation of the theory of probability [5]. The problem arose to them by a gambler. The problem was, Two players of equal skill want to leave the table before finishing their game. Their scores and the number of points which constitute the game begin given, it is desired to find in what proportion they should divide the stakes" [5]. Both Fermat and Pascal worked on this problem and came up with the same conclusion using different proofs [5]. Unfortunately for the math world, Pascal changed his focus of studies in 1654 [1]. This happened after he experienced a life-altering accident. He was driving a four-in-hand

14 PASCAL S TRIANGLE 3 carriage when the horses ran off. Pascal was saved, however, because of the brakes on the carriage. He saw this as a sign from God [5]. He was from that moment on focused on religion and stopped writing papers on math or physics. He began publishing anonymous papers on religious concepts. They were called the Provincial Letters and there were 18 of them. These papers were written to help protect his friend who was in trouble for some of his religious work [1]. While on his religious quest to spread Christianity in 1658, Pascal started to have trouble sleeping because of a toothache. He began to start thinking about math again and this actually stopped him from having pain in his tooth. He saw this as a sign from God that he should continue to think about math [5]. For the next eight days, he worked on the geometry on the cycloid and published a paper on it [5]. This was Pascal s last mathematical work. A few years later he passed away in 1662 at the age of 39. He died from an intense pain after a malignant growth in his stomach spread to the brain" [1]. The Arithmetic Triangle: As stated before Pascal wrote this document in 1954 while working with Fermat. His work did not get published until after his death in 1665 [2]. Looking below we see the arithmetic triangle Pascal created [2]. The lines are put in a ranking system from a certain point. In Figure 1 that point is point G. There are some properties of this triangle that we need to be able to understand the consequences we want to draw from it [2]. Each square block is a cell. Each diagonal line is called a base. The numbers going across the top and down the left sides of the cells are the exponents of the lines. We see that the bases match up with the exponents. The cells in the first row are considered to be in the same parallel rank [2]. It follows the same pattern for each row. The exponent on the left of the row is considered to be its parallel rank. The first column is all in the same perpendicular rank. This also is true as you move through each column. The number on the top of the column is consider the perpendicular rank. All points on the diagonal lines are considered to be on the same base [2]. There

15 4 KEVIN SNYDER Figure 1. Arithmetic Triangle are also reciprocals on this graph. Two examples of reciprocals are the points {E,R} and {M,S} [2]. The difference is that one cell parallel rank is the other cell perpendicular rank and vice versa. We also need to know that the cells that are considered cells of the divide are the cells that have the same parallel and perpendicular rank. The last thing we need to be able to understand is that if you add the rank of the perpendicular and parallel together it will be higher then the rank of the base it is on. For example at point F, the parallel rank is four and the perpendicular rank is three. The bases touch the six exponent. We know 7 > 6, so the idea is true for this instance and it will be true no matter which point we look at [2]. The way to determine the numbers in each of the cells is a fairly easy process. The number of each cell is equal to the value in the cell previous to this cell in the perpendicular rank plus the value in the cell previous to this cell in the parallel rank. Since these values always seem to rely on a previous value, there is only one value that can change. The rest

16 PASCAL S TRIANGLE 5 are set in stone due to this rule [2]. Since the triangle only needs one value to be able to determine the rest of the values we call this triangle first cell a generator [2]. Consequences: From this information Pascal was able to come up with nineteen consequences that always apply to this type of triangle. They are listed below with an explanation for each that it is needed for [2]. For each of the consequences they pertain to all arithmetic triangles. 1. All the cells of the first parallel rank and of the first perpendicular rank are equal to the generator. This just says for our example since our generator is 1 this means all the cells in the first row and first column are equal to 1. 2.Each cell is equal to the sum of all the cells of the preceding parallel rank, comprehended from its perpendicular rank to the first inclusively. This was shown above by how we obtain the value in each cell but it is more saying it is the sum of all of them. For example, a cell in the fifth parallel rank is found by using the fourth parallel rank which is found by using the third rank and so on and so forth. 3. Each cell equals the sum of all cells of the preceding perpendicular rank, comprehended from its parallel rank to the first inclusively. This is the same as above except instead of parallel it is perpendicular. 4. Each cell diminished by unity is equal to the sum of all cells which are comprehended between its parallel rank and its perpendicular rank. This is combining the the second and third consequence. 5. Each cell is equal to its reciprocal. 6. A parallel rank and a perpendicular which have one same exponent are composed of cells all equals the ones to the others. This is true because when this case happens the cells are reciprocals. 7. The sum of the cells of each base is double the cells of the base preceding. Looking at our example the sum of the third base is 4. The sum of our fourth base is 8 which is twice the amount of the third base.

17 6 KEVIN SNYDER 8. The sum of the cells of each base is a number of the double progression which begins with the unit of which the exponent is the same as the base. This is similar to number seven except it says you start from the generator and the pattern will always continue. Since the base is 1 this is true. 9. Each base diminished by unity is equal to the sum of all the preceding. This just says we can get a sum of a base and divide it by 2 and get the sum of the previous base. This is only true if the base is one. If it was something else then we would need to say diminished by the generator. 10. The sum of as many contiguous cells as one will wish from its base, beginning with an extremity, is equal to as many cells of the preceding base, plus again as many except one. This is best shown by an example. If we looked at the third base which contains 1,3,3,1 the sum of that is 8. Suppose we only want the first three so 1,3,3 which is 7. This says that if we look at the base above we get the same sum by taking 2 (1) + 2 (2) + 1 which in turn is true. This will always be true for any continuous cells on the same base. 11. Each cell of the divide is double of that which precedes it in its perpendicular or parallel rank. For instance the divide that contains a parallel and perpendicular rank of 4 is equal to 20. If we look at the cell with the same perpendicular rank but the previous parallel rank it is equal to 10. We see the same thing for the previous perpendicular rank and the same parallel rank. 12. Two contiguous cells being in one same base, the superior is to the inferior as the number of cells from the superior to the top of the base to the number of cells from the inferior to the bottom inclusively. The best way to understand this is by looking at the figure. The expression E is to C as 2 is to 3 is true. E has two cells below it on the same base while C has three cells above it on the same base. We could consider C superior to E in this case. 13. Two contiguous cells being in the same perpendicular rank, the inferior is

18 PASCAL S TRIANGLE 7 to the superior as the exponent of the base of this superior to the exponent of its parallel rank. Again an example from the diagram is helpful. We will say F is to C as 5 is to 3. F is the inferior and C is the superior. 5 is the exponent of the base C and 3 is the parallel rank of C. 14. Two contiguous cells being in the same parallel rank, the greatest is to the preceding as the exponent of the base of that preceding to the exponent of its perpendicular rank. Looking at the diagram, we can say F is to E as 5 is to 2. F is the greatest and E is the preceding. 5 is the exponent of the base E and 2 is the exponent of the perpendicular rank of E. 15. The sum of the cells of any parallel rank is to the last of this rank as the exponent of the triangle is to the exponent of the rank. This says we can take any triangle we want containing a constant base and apply Consequence Any parallel rank is to the inferior rank as the exponent of the inferior rank to the number of its cells. Let s use an example to help show this one. We want to show that F is to M as 4 is to 2 from Consequence 12. If we take the sum of the parallel ranks preceding F we get A + B + C which is 10. If we do the same for M we get D + E which is 5. Now using the Consequence 12, we know this is true since they are on the same base. 17. Any cell that is added to all cells of its perpendicular rank, is to the same cell added to all the cells of its parallel rank, as the number of cells taken in each rank. The way to do this idea is really combining both Consequence 12 and 13 for any cell. 18. Two parallel ranks equally distant from the extremities are between them as the number of their cells. This shows that if you make a triangle with one an edge being the base we will see that the amount of cells in the parallel rank is the same as the amount of cells in the same perpendicular rank. 19. Two contiguous cells being in the divide, the inferior is to the superior taken

19 8 KEVIN SNYDER four times, as the exponent of the base of that superior to a number greater by the unit. This is pretty extensive but it uses a lot of the previous consequences to prove this. Using these consequences allowed Pascal to be able to identify the number in a cell without using the arithmetic triangle but having its parallel rank and perpendicular rank [2]. Ways to Use the Triangle: Pascal came up with many uses for the arithmetic triangle. One was for numeric orders. He came up with orders for the numbers in a certain parallel rank. The first order consist of the row 1,1,1,1,1,1,1,. The second order is the row 1,2,3,4,5,6,. The third order is the row 1,3,6,10,. The pattern continues through out all of the parallel ranks [2]. He notice that the numbers align in an interesting fashion. The first order are just the unit 1. The second order contains all the natural numbers. The third order contains all the triangular numbers and the fourth order contains all the pyramidal numbers. This pattern continues for all the orders. Another idea he related the arithmetic triangle to was combinations. He showed that the cells of any parallel rank equals the number of combinations of the exponent of the rank in the exponent of the triangle. To show this we can look at any triangle. I am going to choose the one that contains the fourth base. I am going to look at the sum of the second parallel. This contains the sum of Now since we have the fourth exponent and the second parallel, the idea says that (4 choose 2) should equal the previous sum. This turns out to be true. He also went into other ideas pertaining to combinations. Pascal also talked about using the arithmetic triangle to determine the divisions in a two player game with deciding who should play and win each game. This was related to the work he did with Fermat [2].

20 PASCAL S TRIANGLE 9 The final use that Pascal talked about was the idea of using the lines of the bases to come up with relations to binomial equations. He found for example that the coefficients of a binomial equation to the fourth power had the same coefficient ratio as the fifth base. An example of this would be 1 A A A A + 1. He later went into ideas pertaining to decimals and negative numbers. This information was useful to Newton as he used this paper to help him come up with a general formula [2]. Other Related Ideas: Since Pascal s passing other scholars have looked at the triangle and found interesting ideas that are related to the triangle. These ideas pertain to the triangle in an actual triangle shape now at the top. One idea is that each item in the triangle is a combination of the (row -1) choose the item number in that row(the item number starts at 0) [4]. For example, in the fourth row, the second item is 3. This equals ((4-1) choose 1). (3 choose 1) does equal three and this works. We also can also see when the triangle is put this way it is symmetrical down the center [4]. Another observation can be made that the sum of each row is 2 n where n =(row-1). For example the sum of the fourth row is = 8,and 2 3 = 8 [4]. Another observation is that if you turn the triangle like a right triangle where one is at the top and then the next row is written horizontally and this carries forever. Now if you take the sums of the diagonals in this triangle you will find the sum of each diagonal is the Fibonacci number for the number of the diagonal [3]. Overall Pascal triangle is more then just a triangle. The applications this triangle can do is off the charts and really handy in a lot of subject areas. Pascal did a great job with his paper and was a brilliant man. References [1] O Connor, J. J., and Robertson, E. F. (n.d.). Pascal biography. MacTutor History of Mathematics. Retrieved November 28, 2012, from http : //www history.mcs.st andrews.ac.uk/biographies/p ascal.html

21 10 KEVIN SNYDER [2] Pascal B. (1654). Treatise on the Arithmetic Triangle. Oeuvres Completes de Blaise Pascal, I-III, Retrieved November 30th, 2012, from http : //cerebro.xu.edu/math/sources/p ascal/sources/arith t riangle.pdf [3] "Patterns in Pascal s Triangle." Interactive Mathematics Miscellany and Puzzles. N.p., n.d. Web. 13 Dec < http : // the knot.org/arithmetic/combinatorics/p ascalt rianglep roperties.shtml >. [4] Stenson, Cathy. "Pascal s Triangle." Combinatorics. Juniata College. Brumbaugh Academic Center, Huntingdon. 31 Aug Class lecture. [5] Wilkins, D. (n.d.). Blaise Pascal ( ). School of Mathematics : Trinity College Dublin, The University of Dublin, Ireland. Retrieved November 25, 2012, from http : // eople/p ascal/rouseball/rb P ascal.html

22 Amy Ankney History Of Math Dr. John Bukowski 13 December 2012 Origins of Probability Theory In a world full of chances, it is a surprise that humans took so long to quantify them into mathematical models and formulas. It took some of the greatest mathematicians in the history of the field to discover the concepts that hid underneath some of the simplest occurrences of life and could even be used to predict the future. The idea of chance and games of chance existed far back into the ancient world. They would take the heel bones from sheep and use them similarly to how we would use dice today. Astragali, as they are called, were used by many oracles in ancient civilizations to make their predictions. They had certain outcomes that represented opinions from gods. Dice made out of clay have been found in Egyptian tombs dated from up to 2000 BC. By the Greek age, they were learning how to cheat by creating loaded dice. However they still did not attempt to learn the math behind their gambles. The earliest documented beginnings of investigating probabilities mathematically did not emerge until the 15 th and 16 th centuries. So that means over 3000 years passed without anyone trying to make some probability abstractions. Most likely this is because in the Greek age, the mathematical advances were by philosophers who logically explained things they understood. They had yet to develop an idea of experimentation that was required to observe any probabilityrelated mathematical patterns. However while the Greeks and Romans believed in the idea of chance, the field of probability was kept dormant even longer by the rise of Christianity.

23 In the scope of Christianity in this era, every random event, even down to a role of the dice, was directly influenced by the intervention of God. The fear of being labeled a heretic by the church dissuaded anyone who possessed the intellectual prowess to develop or publish any calculations in probability. There may not have been publications directly on probability, but there are documented Greek, Chinese, and Arab mathematicians who attempted to calculate combinations. Yet no one put those ideas with outcomes of a random event together until the end of the dark ages. Out of the ashes of the dark ages rose many developments in a range of fields, including probability. In the late 15 th century, Luca Pacioli proposed the question: A and B are playing a fair game of Balla. They agree to continue until one has won six rounds; The game actually stops when A has won five and B three. How should the stakes be divided? As the man who concluded that the solution of the cubic was impossible, Luca Pacioli was also incorrect in his solution to the problem of points. He believed the stakes should be divided 5:3, but this does not take into account the actual outcome of any subsequent rounds, so it was incorrect. This is the question that would be tossed around from mathematician to mathematician throughout this time of intellectual prosperity. Although his work was incorrect, he was the first to document that style of conjecture. Unlike much of his work, this is believed to be originally his own advancement. Around the same time, one of the most bizarre characters in the history of math was hypothesizing about probability. Gerolamo Cardano is a man who claims that he was torn from his mother s womb at birth and that he is of close descendants from giants. He is most notoriously known for having published the solution of the cubic after having been given it in confidence from a fellow mathematician and friend, then receiving most of the credit for the

24 work. He attended University of Pavia where he studied medicine. However, in this time during and after college, he was known for having a rampant gambling addiction. Unsurprisingly this is where his interest in probability was born. He sought to create a model that illustrated the outcome of a random event. This led him to the discovery that if there are m desired equally likely events of n possible outcomes, then the probability of the desired outcomes is m/n. This is now considered the classical definition of probability and the first documented idea of theoretical probability. Although his work was written in 1525, it was not actually published until 1663, when the attention had moved to two other well-known mathematicians and their developments in the field of probability. Chevalier de Mere, a well-known mathematician and avid gambler, proposed the problem contemplated by Pacioli to Blaise Pascal of how to fairly split up a stake between two gamblers whose game had been interrupted before they finished. He was so intrigued that he sent a letter to a fellow mathematician Pierre Fermat. Fermat thought highly of Blaise Pascal because when a controversy arose of Descartes criticizing Fermat s method of finding maxima and minima, Blaise s father, Etienne Pascal, was one of the numerous mathematicians who came to Fermat s defense. Born on August 17 th, 1601, in Beaumont-de-Lomagne, France, Fermat, although not a mathematician by trade, was one of the greatest mathematical minds of this era. He was educated at a local monastery and then studied law at the University of Toulouse. As a counselor of the Parliament, when courts were at recess he was expected to keep distant from his fellow citizens. This allowed for a lot of personal time for intellectual pursuits, and was a perfect environment for his mathematical genius to flourish. He was a bit of a recluse and did his best work in isolation. He rarely published anything because he refused to put his work in a polished form. He

25 is most noted for Fermat s Last Theorem, where he scribbled in the margin of a book that he had a proof for Pythagorean triples to power greater than two. However, he claimed he didn t have space for his proof in the margin and left it at that. He never made any publications on probability, but his correspondence with Blaise Pascal is attributed to synthesizing the ideas that would become the groundwork for the field of probability. Blaise Pascal, 22 years Fermat s junior, was a mathematician beyond his years. His father had always pushed his reasoning skills rather than his memory. He decided that he would remove all works of math from his home until Blaise was 15. But at the age of 12, Blaise was found to be sketching out mathematical diagrams and had deduced Euclid s propositions on his own. By 19 he had invented the first digital calculator, a technology that wouldn t be matched until the 1940 s. His health declined at this time so he had to stop working. He would battle with bouts of migraines and eventually stomach cancer for the rest of his life. His illnesses interrupted his work and prevented him from fulfilling his full mathematical potential. Blaise was also known for being very deeply religious. However, unlike those during the rise of Christianity, he used probability to rationalize his religious beliefs. His famous quote was: If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing. The initial letter from Pascal to Fermat was never found, but the subsequent correspondence has been translated, and is attributed to laying the foundation for the subsequent work in probability afterward. Fermat proposed that if he was dividing the winnings between two men, after one roll of the dice, then the forfeiting opponent should take 1/6 of the total. Then to divide the winnings on a proposed second roll of the dice he would take 1/6 of the 5/6 left from the first roll, leaving him

26 with 5/36 of the original total. Then, preceding the same for the third, fourth, and fifth roll. But they differed in their idea of how to split up the winnings when you have thrown the dice three times and failed to get a six, but on your fourth turn you agree to abandon the attempt and just take part of the wager. Fermat claimed that Pascal calculated 125/1296 would be a fair cut to give the withdrawing opponent, because Pascal based it off of the idea that both players were agreeing to not take the last throw. Fermat believed that the opponent should get 1/6th of the total wager because each throw has the same chance of getting the desired 6. In Pascal s response letter he admitted his error and moved on to his next attempt at the solution. Pascal sets up the problem as having two Players, A and B, who have wagered the same amount and decided to play until one has won n amount of games. Then they decide to end the game when Player A is one game away from winning and Player B is two games away from winning. Player A has a points and player B has b points and there are two more rounds. Then how should they divide the wager? Pascal goes on to solve it by generalizing that player A has n- 1 games and player B has n-2 games. So Pascal sees it as a straightforward way to divide the stakes since there are 4 possible outcomes: Either Player A wins the next round, and wins the second round. Player B wins the first round and Player A wins the second. Player A wins the first round Player B wins the second round. Player B wins the first round and the second round. So the first three options result in Player A winning over all and Player B wins over all in the fourth option. So Pascal concludes that if they didn t play the rounds, Player A should get 3/4 of the wagers and Player B gets 1/4. However this is wrong. Fermat continues to argue that there are 3 possible outcomes, not four. The problem is that the frequencies of the three or four outcomes are not equal in either case. Both Fermat and Pascal struggled to wrap their heads around this. So Pascal created a table to lay out all the possible ways to divvy up 4 points to two people. The

27 following table represents that by a being a point to player A and b being a point to player B. Then the winner of each is shown by a one or a two. As illustrated in the table, the actual chances of player B winning are 5:11. a a a a a a a a b b b b b b b b a a a a b b b b a a a a b b b b a a b b a a b b a a b b a a b b a b a b a b a b a b a b a b a b Then Pascal moves on to illustrating his short cut for fairly divvying up wagers from any game. His solution to the problem is a complicated method of combinatorial propositions that are referenced in his Treatise on the Arithmetic Triangle, now known as Pascal s Triangle. His general proof for finding the value of the first game of any number of attempts is as follows: Let the given number of games be, for example, 8. Take the first eight even numbers and the first eight odd numbers thus : 2, 4, 6, 8, 10, 12, 14, 16 and 1, 3, 5, 7, 9, 11, 13, 15. Multiply the even numbers in the following way: the first by the second, the product by the third, the product by the fourth etc; multiply the odd numbers in the same way; the first by the second, the product by the third, etc. The last product of the even numbers is the denominator and the last product of the odd numbers is the numerator of the fraction which expresses the value of the first one of eight games (David p 233). This translates into a much more familiar looking equation of. The response to Pascal s methods has also been lost, but Fermat proposed a solution where Player A needs n-a games and player B needs n-b games. So the game will be decided

28 within 2n-a-b-1 rounds. Then he created all the possible sequences of rounds that would result in a win for Player A and then all the sequences of rounds that would result in Player B winning. However Pascal pointed out that his error was in not acknowledging that the game may end in less than 2n-a-b-1 rounds. So he did not have all the possible sequences, since he did not even consider the ones less than 2n-a-b-1. In response, Fermat affirmed Pascal s solution and conceded his error. Pascal never replied to this letter. Fermat sent another letter that explained in length his belief that there was no harm in still considering the sequences that may not actually happen since it made the enumeration of outcomes simpler. Although the letters are considered the beginning of probability, the word probability was not used in any of the letters exchanged. Instead, they spoke of hazards and what hazards they were taking. There are critics that say this should not be considered the birth of probability, but rather considered great bounds in the fields of combinatorics and enumeration. However, credit is due for creating the ideas that snowballed further into the field of probability. Over 700 miles away, Christiaan Huygens was working on his publications on astronomy and optics in the Netherlands. When he presented his discoveries of Saturn s first moon in Paris, he learned of the correspondence on probability between Fermat and Pascal. This piqued his interest and upon returning home he delved into the calculus of probability. He sent off his findings on probabilities related to dice games to numerous French mathematicians hoping that his methods were on the right track. Through the network of mathematicians he corresponded with in France, he was sent the same problem of points question that was posed to Fermat and Pascal. Huygens sent his solution back within two weeks of receiving the problem. He received a letter in return saying he had the same solution as Pascal. So Huygens went back to the problem of points and solved it again. This became the eleventh proposition in his book of fourteen. He

29 titled the book, De Ratiociniis in Ludo Aleae (Calculations in Games of Chance). It was published only three years after his initial visit to Paris that started him on his studies. The fourteen propositions were ground-breaking at this time, although quite different from today s knowledge of probability. Still, it was far more comprehensive than the discoveries Fermat and Pascal had made and was used as the textbook of probability for over 50 years. It wouldn t be until Bernoulli published his work, Ars Conjectandi, that the field of probability would change, but Huygens still holds unchallengeable credit for having hit the tip of the iceberg in the field of probability. Jacob Bernoulli, one of the famous Bernoulli brothers, originally attended the University of Basel in Germany for theology. However, like many mathematicians before him, when he was 18 he became intoxicated with the wonder of math after studying Euclid s Elements. His focus was on astronomy, so in the time between when he traveled to study stars, he spent his time teaching and researching. As a student, he read Huygens s publication and that sparked his interest in the subject of probability. Most of his discoveries were in the field of infinite number theory. So it doesn t come as a surprise that his theories in probability were heavily tied into infinite number theory. In his major publication on probability, Ars Conjectandi, he deduced one of the most ground-breaking ideas of this era for probability. He found that the larger number of trials you have, the closer outcomes approach the expected value. For example, suppose there are 3 white balls and 2 black balls in an urn. By the time 25,500 trials are reached, the chances of getting a white ball outside of 29/50 and 31/50 is only 1/1000. In other words, to see a proportion outside of the expected 3/5 white to 2/5 black is about a 1/1000 chance. This is the first published work that acknowledges the idea that probability becomes closer to certainty as

30 the number of trials approaches infinity. He insightfully expanded this idea to the existential question of fate. If thus all events through all eternity could be repeated, by which we would go from probability to certainty, one would find that everything in the world happens from definite causes and according to definite rules, and that we would be forced to assume among the most apparently fortuitous things a certain necessity, or, so to say, FATE (David p 137). Unaware to Bernoulli, this view on probability would lay a rift between statisticians for centuries to follow. This is considered the frequentist view of probability, as opposed to Bayesian. Bernoulli uses his argument to imply that certainty is something that can be attained, while the Bayesians put a stronger emphasis on the uncertainty quantified by probability. It boils down to the attitude toward creating intervals. A frequentist wants to have a large enough number of trials so they can say that the mean is in the interval or it is not in the interval. However the Bayesian will look at the interval being based on the data, rather than the population, so they will imply the uncertainty by saying: there is a 95% probability that this interval contains the mean. It seems like a very insignificant difference in opinions, but it is two entirely different understandings on the applications of statistics. Another Mathematician inspired by Huygens s publication De ratiociniis in ludo aleae was Abraham de Moivre. Little is known about his early life. In 1678, he began at the Protestant College in Sedan at the impressive age of eleven, however the Protestant reform movement was expanding and forced him to leave and attend the college in Saumur, where he first studied Huygens s work. He did not receive any official mathematical training until moving to Paris where he received private lessons. Then at the pinnacle of the religious counter-reformation, he

31 was imprisoned for being a Protestant. It is unclear how long he was detained; some sources say he was released shortly, while others say he was imprisoned up to almost three years. Regardless, by the time he was released he was fully versed in the classic works. He traveled to England to escape the religious turmoil of France. He worked as a tutor and spent his free time, even the time travelling from pupil to pupil, studying any mathematical work he could get his hands on and networking with English mathematicians. Also at this time, he began to go to a coffee house after his work tutoring, where he would charge gamblers for calculating their odds. In 1718 he published his first work on probability, The Doctrine of Chance. In this work he expands on Bernoulli s law of large numbers. He proposes the idea that although Bernoulli acknowledged the idea, he did not explicitly state what he alluded to is standard deviation, which decreases as the number of trials increases, and is represented in the formula where n is the number of trials, p is the number of desired outcomes, and q is (p-1). This is now known as the standard deviation of the binomial distribution. He then went on to apply this to the binomial theorem, that as n increases, a binomial curve begins to look like a normal bell shaped curve. This is known as the de Moivre-LaPlace Theorem. Pierre Simon Laplace, another French mathematician, also expanded on Bernoulli s idea of large numbers and deduced the same idea of de Moivre. It came almost 50 years later, but de Moivre s normal law was not noticed in his work until the late 1800 s. So the theorem has been dubbed the de Moivre-Laplace Theorem to give credit to both mathematicians who found it. In the introduction to his publication on probability, Théorie Analytique des Probabilités, Laplace says: The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which oft times they are unable to account (David 410). Laplace has been known for his

32 inductive reasoning in probability, which typical of Bayesian probability, was very groundbreaking for this time. With his outstanding reasoning abilities, he developed the method of least squares; however he did not publish it with any relation to probability. Laplace is the last of the mathematicians to make great bounds in the field of probability during this time period in France. It is the close of an intellectual era that was the perfect climate for mathematical networking, building off of one another s ideas and findings. The history of probability theory could go on infinitely to cover more mathematicians before and during that time, and the mathematicians still making progress today. There are numerous mathematicians of the current era that made bounds in the field but the ones discussed previously are given the most glory for establishing the ideas that mathematicians would draw on for centuries. They may not have been totally accurate, but they were the first to quantify the random events of life. What was previously referred to as righteous intervention from God had been discovered to be relatively predictable events.

33 Works Cited Abrams, Bill. "A Brief History of Probability." Second Moment, Web. 07 Dec Annis, Charles, PE. "Frequentists and Bayesians." Frequentists and Bayesians. N.p., 15 Jan Web. 07 Dec Apostal, Tom M. "A Short History of Probability." Caclulus. 2nd ed. Vol. 2. N.p.: John Wiley & Sons, N. pag. A Short History of Probability. Web. 07 Dec David, F. N. Games, Gods and Gambling: The Origins and History of Probability and Statistical Ideas from the Earliest times to the Newtonian Era. New York: Hafner Pub., Print. Devlin, Keith J. The Unfinished Game: Pascal, Fermat, and the Seventeenth-century Letter That Made the World Modern. New York: Basic, Print. O'Connor, J. J., and E. F. Robertson. "Blaise Pascal." Pascal Biography. N.p., n.d. Web. 07 Dec "Sources in the History of Probability and Statistics." Sources in the History of Probability and Statistics. Xavier University, n.d. Web. 07 Dec

34 JAMES STIRLIn!G An!D N!UMBERS OF THE FIRST An!D SECOn!D KIn!D MATT JOHA(N! 2 ) Many know James Stirling as a British architect who built the Florey Building at Oxford University in The James Stirling to be considered is not an architect, but a mathematician who studied at Oxford University in The mathematician James Stirling has an important role in some breakthrough periods for mathematics. He lived an interesting life, meeting people such as Isaac Newton and Nicolaus Bernoulli I. Stirling solved many mathematical questions in his lifetime and published some very important works. Stirling s mother and father were Archibald Stirling and Archibald s second wife, Anna Hamilton. Archibald and Anna had James in May of 1692 in Garden, Scotland. James was born into a very supportive family to the Jacobite cause. Many did not agree with the supporters of the Jacobite cause and this showed starting at the age of 17. Stirlings father, Archibald, was arrested after accusations of high treason because of his Jacobite support. [1] Jacobitism is the movement in Great Britain and Ireland between 1688 and 1746 to restore power to King James II of England. [2] Not much is known about Stirling s younger years. The first known information about Stirling is that he traveled to Oxford in the fall of [1] He went to Oxford to matriculate at a University. Matriculate comes from the Latin word matricula, or little list. [3] In January of 1711, Stirling s plan came true when he matriculated at Balliol College Oxford. There is no certainty, but it is said that Stirling also studied at the University of Glasgow. [1] Date: December 13,

35 2 MATT JOHA(N! 2 ) Stirling was awarded a scholarship in 1711 with only one rule, to swear an oath when matriculating. His Jacobite sympathies would not let him do so, but he was excused. In 1715, there was a Jacobite rebellion, which created a problem for Stirling. Stirling was withdrawn from his excuse to not swear on the oath. Upon refusal of swearing on the oath, Stirling lost all of his scholarships and could not graduate from Oxford. [1] Stirling s support for the Jacobites now hindered his life in the biggest way possible. Stirling may have bounced around in Oxford for a while, but that ended when he left for Italy.[4] While in Italy Stirling almost became a professor of mathematics in Venice, but for unknown reasons this did not happen. All was not lost, however. In 1717, while in Italy, Stirling published his first work. The work titled Lineae Tertii Ordinis Neutonianae extends ideas of other mathematicians. There are results on the curve of quickest descent, the catenary and orthogonal trajectories. The problem of orthogonal trajectories was first raised by Gottfried Wilhelm Leibniz. [1] Stirling is known to be the mathematician who solved the problem in Some other famous mathematicians working on the problem as well include Johann Bernoulli, Nicolaus Bernoulli I, Nicolaus Bernoulli II and Leonard Euler. In 1718, Stirling published more work, this time through Newton a paper titled Methodus Differentialis Newtoniana Illustrata or the Illustrated Newtonian Differential Method in English.[4] In 1719, Stirling decided to submit this to the Royal Society of London from Venice. The paper was received and reported to their meeting on June 18 th of the year. The years from 1716 to 1719 had been a busy time for Stirling, but it seems that he would not slow down. In 1721 Stirling was in Padua where he took classes under the chair of the great Nicolaus Bernoulli II. It wasnt long before Stirling returned to Glasgow. This was at a similar time when Nicolaus Bernoulli II left Padua. [1] In 1722, Stirling left with the intentions of becoming a teacher in London. In 1724 Stirling travelled to London. He stayed in London for 10 years and was very active in the mathematics world. Stirling corresponded with many mathematicians and

36 JAMES STIRLIn!G An!D N!UMBERS OF THE FIRST An!D SECOn!D KIn!D 3 had a good friendship with Isaac Newton. Newton proposed Stirling to the Royal Society of London for his work. In 1726 Stirling was elected for the society. Things were going well for Stirling at this time and in 1727 he reached his goal and became a teacher at William Watts Academy on Little Tower Street, London. [1] The year 1730 proved to be Stirling s most important, this was the year that Stirling published the Methodus Differentialis. The title is translated in English to mean The Method of Differences. In this book is treatise on infinite series, summation, interpolation and quadrature. Also in this book is the asymptotic formula for n!, which made Stirling famous and why he is relevant today. It was Proposition 28, example 2 of Methodus Differentialis that approximated n!. The approximation is n! 2πn( n e )n and appropriately so called the Stirling approximation.[5] Abraham de Moivre also published Miscellanea Analytica in 1730 deriving the series expansion formula. [1][5] Stirling wrote de Moivre a letter pointing out the errors that he had made. Stirling s approximation of n! is often thought of as ln n! = n ln n n.[8] Writing the expansion this way makes it easier to compare graphically. Figure 1. ln n! = n ln n n

37 4 MATT JOHA(N! 2 ) The graph of ln n! and n ln n n shows that Stirling s approximation of n! is acceptable. The approximation does show pretty poor comparisons, but as n increases the two converge closer and closer. In Methodus Differentialis Stirling expands on the Newton series. A Newton series is denoted as P 0 (z) = 1, P 1 (z) = z, P 2 (z) = z(z 1), P 3 (z) = z(z 1)(z 2),, P k (z) = z(z 1) (z k + 1). The Newton series can also be written as, f(z) = a k z(z 1)(z 2) (z k + 1) k=0 = a 0 + a 1 z + a 2 z(z 1) + a 3 z(z 1)(z 2) + [4]. At the beginning of his book Stirling studied the coefficients A m n in z m = A m 1 z + A m 2 z(z 1) + A m 3 z(z 1)(z 2) + + A m n z(z 1) (z m + 1). Stirling obtained the following for A m n : z = z z 2 = z + z(z 1) z 3 = z + 3z(z 1) + z(z 1)(z 2) z 4 = z + 7z(z 1) + 6z(z 1)(z 2) + z(z 1)(z 2)(z 3) The coefficients were written down in Stirling s first table. This table represents what we now call the Stirling numbers of the second kind.

38 JAMES STIRLIn!G An!D N!UMBERS OF THE FIRST An!D SECOn!D KIn!D 5 Figure 2. Stirling Numbers of the Second Kind n The Stirling numbers of the second kind are denoted as S(n, k) or. The Stirling k numbers of the second kind are the number of ways to partition a set of n objects into k non-empty subsets. [4] A good way to think about creating these partitions is by having number of sweaters be n and having number of boxes be k. It is the summer and one looks to put the sweaters away into boxes. The sweaters must all go into a box, but no box may be left empty. If one has 3 sweaters and 3 boxes, there is only one way to put the sweaters into boxes and that is to put one sweater in each box. Similarly, if there are 3 sweaters but only 1 box to put them in, there is only one way to put them into boxes. There is only one way to put them into the boxes and that is to dump all 3 sweaters into the one box. If one has 3 sweaters and only 2 boxes to put them into is where partitioning gets fun. There are three sweaters, one of which is green, another that is purple and another that is red. Now the green and red sweater can be in separate boxes. Then either the purple sweater is in the box with the green sweater or in the box with the red sweater. If the green sweater and purple sweater were our starting sweaters in separate boxes, then the red sweater can either go in the box with the green one or in the box with the purple one. Notice that there had already been a box that was accounted for with a purple sweater

39 6 MATT JOHA(N! 2 ) and red sweater. There are no more ways to do this and there are only three ways to put these three sweaters into 2 boxes. [7] Notice, from the Stirling chart, the Stirling number S(3, 1) = 1 and the Stirling numbers S(3, 2) = 3 and S(3, 3) = 1. Looking ahead on Stirling s table one can say that there are 7770 ways to put 9 sweaters into 4 boxes (S(9, 4)). While the Stirling numbers of the second kind originally came from coefficients of the Newton series of z k, there are many helpful uses of these numbers and why the Stirling numbers are still interesting to study today. Also in Stirling s Methodus Differentialis is Stirling s study on interpolation. In particular, Stirling studied the sequence T n+1 = nt n with T 1 = 1. The numbers are as follows: T 1 = 1 = 1 T 2 = 1(1) = 1 T 3 = 2(1) = 2 T 4 = 3(2) = 6 T 5 = 4(6) = 24 T 6 = 5(24) = 120 T 7 = 6(120) = 720 T 8 = 7(720) = 5040 These numbers are the start of what are now known as the Stirling numbers of the first kind. The Stirling numbers of the first kind are denoted as s(n, k) or n. The Stirling k numbers of the first kind are known as the number of permutations of n elements with k disjoint cycles.[4]

40 JAMES STIRLIn!G An!D N!UMBERS OF THE FIRST An!D SECOn!D KIn!D 7 Using the example of sweaters, the Stirling numbers of the first kind can also be described. It is now the winter and there is an interest of putting the sweaters into the closet. In the closet there are only circular racks. Stirling numbers of the first kind are the number of ways to put n sweaters onto k circular racks. This means that there is now a concern about order, where in the summer one just throws them into the boxes and puts them away. One can now order the sweaters possibly in weekday order. [7] The number of way to put 3 sweaters onto 1 circular rack is not one. There are purple, red and green sweaters again. Now that order matters there can be pgr or prg where p =purple sweater, r =red sweater and g =green sweater. Rotating the sweaters does not change anything so pgr = grp = rpg. There are 2 ways to put 3 sweaters on 1 circular rack.[7] This is also known as s(3, 1) = 2. The Stirling numbers of the first kind are as follows: Figure 3. Stirling Numbers of the First Kind

41 8 MATT JOHA(N! 2 ) Stirling s series of T n+1 became known to be the Stirling number of the first kind when k = 1. The formula s(n + 1, k) = s(n, k 1) ns(n, k) helps understand the rest of the table. Stirling published some great work in 1730 and the work did not go unnoticed. In 1736 Leonard Euler wrote Stirling a letter. The great Euler was impressed with what Stirling had published and wanted to learn more. Stirling was extremely busy at this time in his life and took two years to respond to the letter. In Stirling s response, he said that he would put Euler s name forward for election to the Royal Society, but never got around to actually doing so. Euler was proposed, by many mathematicians that were not Stirling, to the Royal Society in Stirling was not done publishing work in He published a paper on the ventilation of mine shafts. There was a major rebellion of the Jacobite cause in this year. In 1746 the chair of Edinburgh had died and Stirling was considered to be the new chair. Stirling s strong support for the Jacobite cause made it impossible for him to get the chair. Stirling was elected as a member of the Royal Society of Berlin in It only took seven years for Stirling to resign because he could no longer afford the subscriptions. James Stirling was not an architect, but a very good mathematician. He was associated with names such as Newton, Bernoulli, and Euler, who thought that he was very intelligent. Being a big supporter of the Jacobite cause held him back in multiple ways throughout his life. Even with distractions and hatred Stirling was able to publish many papers, and we are still interested in his work centuries later. It is interesting that his name is not used more often and more well-known. This could be because of his support for the Jacobite cause holding him back from a lot in his life. Stirling died in December of 1770 and his work is still studied today. things. Supporter of the Jacobite cause or not, he did some great

42 JAMES STIRLIn!G An!D N!UMBERS OF THE FIRST An!D SECOn!D KIn!D 9 References [1] James Stirling. Stirling Biography. The MacTutor History of Mathematics archive, n.d. Web. 5 Dec. 2012, [2] The Jacobites, Britain and Europe , Daniel Szechi, Manchester University Press 1994 [3] Matriculate. Merriam-Webster. Merriam-Webster, n.d. Web. 5 Dec. 2012, [4] Boyadzhiev, Khristo N. Close Encounters with the Stirling Numbers of the Second Kind. Mathematics Magazine 85.4(2012): Print. [5] Gale, Thomas. James Stirling Biography. BookRags. BookRags, n.s. Web. 5 Dec [6] Milson, Robert. Stirling numbers of the first kind (version 3). PlanetMath.org, [7] Stenson, C. (2012, October). Stirling Numbers. Speech presented at Juniata College, Huntingdon, PA. [8] Schroeder, Daniel V., An Introduction to Thermal Physics, Addison Wesley, 2000,

43 Seven Bridges of Königsberg: An Eulerian Path Johnson 1 During the time of the nomads of the earliest century of man, mathematics had no presence in the world. As these prehistoric men settled during the Neolithic period, the need for mathematics began to grow. Counting livestock and measuring areas of fields developed into the two major branches of mathematics: arithmetic and geometry, respectively (Dunham). As the years progressed, so too did the field of mathematics. The importance of mathematics in society grew as society and mathematics itself grew more complex. Many fields of mathematics were born to satisfy a particular need of society, usually to make daily life easier for the people. Fascinatingly, however, mathematics did not always originate from a search to alleviate mankind s struggle. Graph theory, which is now known as a subdivision of combinatorics, had a humble, almost silly beginning. Leonhard Euler, renowned for his work with infinitesimal calculus and other fields of mathematics and physics and remembered by the many equations, functions, and theorems that bear his name, discovered the field of graph theory when he published his work about a puzzle in This publication, The Solution of a Problem Relating to the Geometry of Position, laid the groundwork for graph theory, a field of mathematics that would eventually make major contributions in the fields of physics, chemistry, biology, linguistics, computer science, and many more, by examining the well-known puzzle known as the seven bridges of Königsberg. Leonhard Euler s lifelong dedication to the sciences and, in particular, mathematics eventually led to his founding of graph theory. Euler was born on April 15, 1707, in Basel, Switzerland. His father, Paul Euler, was a Protestant minister who taught Leonhard mathematics at a very young age. Even though the school Euler attended when he was young was very poor and lacked a mathematical curriculum, he still managed to tame the passion for mathematics he learned from his father with mathematical readings on his own time. At age 14, Euler was sent to attend school at the University of Basel to prepare for the ministry. It did not take long for his potential in mathematics to be noticed by

44 Seven Bridges of Königsberg: An Eulerian Path Johnson 2 the very famous Johann Bernoulli, a professor at the university. Euler would have a longstanding relationship with the Bernoulli family for the rest of his life. It is Johann Bernoulli who persuaded Euler s father to allow young Leonhard to change his area of study from theology to mathematics. Once on the mathematical path, Euler began to excel in the field. By the time he completed his studies in 1726 at the age of 19, he had a paper in print, a second place finish in the grand prize competition in the Paris Academy, and a job offer for a professorship at the St. Petersburg Academy of Sciences in Russia. He took the job in 1727 and quickly became the senior chair of the mathematics department in 1733, at the age of 26. From that time until his death on September 18, 1783, Euler established himself as the most prolific writer of mathematics of all time (O Connor). Of those writings, a particular article of interest, Solutio problematis ad geometriam situs pertinentis (The Solution of a Problem Relating to the Geometry of Position), was written on a puzzle developed by the people of a city in Eastern Prussia and was the building blocks for the field of graph theory. (O Connor) The Solution of a Problem Relating to the Geometry of Position was based on the understanding of a popular puzzle from Eastern Prussia about the city of Königsberg. Königsberg, now known as Kaliningrad, is broken up by the River Pregolya, formerly known as the River Pregel. The river splits the city, forming an island in the river known as Kneiphof and Figure 1: The seven bridges of Königsberg in the context of the city. dividing into two waterways at the far end of the city. Figure 1 depicts the city. Highlighted are the river and seven bridges. These seven bridges made travel between the different parts of the city simpler for the inhabitants of Königsberg. Throughout the years, the citizens of Königsberg would entertain themselves by attempting to traverse each of the seven bridges only once on their travels. Many believed this to be an impossible task because everyone who

45 Seven Bridges of Königsberg: An Eulerian Path Johnson 3 attempted failed; however, there was no explanation as to why this was not an accomplishable feat. In 1736, a 29-year old Euler decided to look at this puzzle in more detail, and he published his paper on the subject. (Biggs 1-2) In the opening paragraph of his paper, Euler acknowledges that the problem of the seven bridges of Königsberg is of a branch of geometry that receives little attention: the geometry of positions. He mentions that Leibniz first mentioned this branch of geometry that is overshadowed by the geometry concerned with magnitudes. Euler explains that geometry of positions is special because it is concerned only with the determination of position and its properties (Biggs 3) and not with calculations or measurements. Geometry of position, as described by Euler, evolved into what is now known as graph theory. As a result, Euler s paper is known as the first paper to have been written on graph theory. Euler was looking for geometry of position problems to analyze because it had not yet been satisfactorily determined what kind of problems [were] relevant to this geometry of position, or what methods should be used in solving them (Biggs 3). He found the problem he was looking for in the seven bridges of Königsberg puzzle because its solution concerned no calculations or measurements, only position. The aim of his paper was to describe the rules and methods necessary for solving this problem and others like it. (Biggs 3) Euler begins his explanation of solving the Königsberg bridge problem by labeling the components of the city and breaking it down into bridges and landmasses. This is shown in Figure 2 excerpted from Euler s paper. The land masses are labeled A-D, and the Figure 2: The seven bridges of Königsberg as drawn by Euler in The Solution of a Problem Relating to the Geometry of Position.

46 Seven Bridges of Königsberg: An Eulerian Path Johnson 4 bridges are labeled a-g. Because Euler wishes to devise a solution for any problem involving a number of landmasses and bridges, he denies the usefulness of a guess-and-check method of tracing paths until one is found or all paths are exhausted. Instead, Euler reasons that he is to find a way in which A-D can be arranged so that each bridge between two landmasses is crossed only once. He shows that for this particular problem, it will be an eight letter arrangement because between each landmass, AB for example, lays one bridge. Since seven bridges must be crossed, eight letters must be used in total to represent the crosses. Moreover, the total number of times someone can traverse an area (A, B, C, or D) is equal to eight. However, he decides that it is best he finds a rule for finding whether or not an arrangement can exist before finding what the particular arrangement is. (Biggs 4) In order to find a rule, Euler considers one area with a number of bridges leading to it. He shows that if one bridge leads to the area, the area will be traveled only once. If three bridges lead into the area, no matter where the person starts, the area will be traveled twice. If there are five bridges, the area will be traversed thrice and so on. Therefore, Euler reasons that if the number of bridges is any odd number, that number increased by 1 then half is the number of times the area is entered. When he applied this to the seven bridges problem, he discovered that A is entered thrice, and B, C, and D are entered twice each. Since these areas in whole are traveled nine times (represented by nine letters), the seven bridges problem is not possible since he showed earlier that for each bridge to be crossed only once, the total number of area visits had to equal eight. Euler then expands his rule to include an even number of bridges by the same thought process. If two bridges lead to an area, the area can be traversed either once or twice: once if the journey is not started in the area, twice if the journey is started in the area. In the case of four bridges leading to an area, if the journey does not start in the area, the area is traversed twice, and if the journey begins in the area, the area is traversed thrice and so on. Therefore, Euler reasons that if the number of bridges is even, than the number of times an area is traversed is equal to half the number of bridges if the journey does not start in the area and it is equal to

47 Seven Bridges of Königsberg: An Eulerian Path Johnson 5 half the number of bridges plus one if the journey begins in the area. With the new rules, Euler devises a method for solving such problems as the seven bridges problem. (Biggs 4-5) In solving a problem involving areas separated by bridges, Euler lays out a simple method for finding whether or not a path can be found that travels each and every bridge only once. First, he labels the areas with capital letters. Then he takes the total number of bridges and adds one. This number is utilized later. Next he writes the areas in a column and notes the number of bridges leading to each in another column. Then he indicates those areas with an even number of bridges leading to it with an asterisk. Next to the even numbers, he writes half of the number. Next to the odd numbers, he increases the number by one and then halves it. He sums this last column and compares it to the number found at the beginning of the process. If the sum of this column is greater than the number of bridges plus one, the journey is impossible. If the sum is one less or equal to the number of bridges plus one, the journey is possible given that if the sum is equal, the journey begins from an area with an odd number of bridges and if the sum is one less, the journey begins from an area with an even number of bridges. Euler applies this process to Königsberg, and again the result was that it is impossible to travel across the bridges only once in a single journey. After showing this, he goes through the method again with fifteen bridges and six lands. This example was included to show an instance where it was possible to traverse all the bridges only once in a single journey. He concludes his paper by summing his findings. (Biggs 6) In the years after Euler s paper, the graph theory field he started grew exponentially and modern graph theory developed Euler s method of solving the seven bridges of Königsberg into Eulerian path problems. In modern graph theory, an Eulerian path is a path which contains each edge of a graph only once in a sequence of edges and vertices. A graph consists of a finite set of vertices and a finite set of edges (Biggs 9). In comparison to Euler s paper on the subject, not much has changed other than the wording. An Eulerian path is a seven bridges of Königsberg problem. The edges are the bridges and the

48 Seven Bridges of Königsberg: An Eulerian Path Johnson 6 vertices are the areas. Figure 3 shows the seven bridges of Königsberg in graph form. The rules one follows when using the method devised by Euler are still the same as well. In the summary of his paper, Euler reduces his Figure 3: The seven bridges of Königsberg in graph form. rules down to three simple rules that allow him to skip the steps he went through to solve the problem. After observations of his results, Euler noticed that he can determine whether or not a path is traversable by the number of bridges leading to the areas. He concluded that if more than two areas had an odd number of bridges leading to them, the path was impossible to devise. If exactly two areas had an odd number of bridges leading to them, it was possible to traverse if the journey began in one of these two areas. If none of the areas had an odd number of bridges leading to them, the journey can be accomplished from anywhere (Biggs 8). In modern graph theory, the number of bridges leading to an area is termed as the degree of the vertex. If there are an odd number of bridges leading to an area, that vertex has an odd degree. An even number of bridges leading to an area would be an evendegreed vertex. In today s graph theory, it is still true that if there are more than two odd-degreed vertices, there exists no path; if there are exactly two odd-degreed vertices there exists a path that starts on either of those vertices, and if there a no odd-degreed vertices, the path can start anywhere. This last case of no odd-degreed vertexes is now known as an Eulerian circuit because the path starts and ends at the same place ( Eulerian Paths ). The implications of this work done by Euler and those that followed him and helped build the field of graph theory are ubiquitous in today s world. In computer science, graph theory is used to represent the flow of information, networking of communication, organization of data, and computation of devices. In linguistics, it is used to model natural language. In physics, graph theory is used in three-

49 Seven Bridges of Königsberg: An Eulerian Path Johnson 7 dimensional modeling of complicated atom structures. In biology, Eulerian paths are used to describe the migration habits of animals, where the area of habitat is represented by a vertex and the migration paths are represented by edges. The list of practical uses of graph theory continues on and without the contributions of Euler, who knows if these processes used every day in a myriad of fields would be as far along as they are today. ( Graph Theory ) Using the seven bridges of Königsberg puzzle, Euler, in The Solution of a Problem Relating to the Geometry of Position, was able to derive a solution to the first problem to ever to be written about in graph theory. The work that he revealed in the paper has stood the test of time and is now the most basic of concepts for the field of graph theory. The humble origins of graph theory, like the humble origins of Euler, laid the groundwork for importance in the field of mathematics. Although it was not developed to ease the lives of the average citizen, Euler s paper, as well as the resulting field of mathematics, has certainly left an impact on the world we live in today.

50 Seven Bridges of Königsberg: An Eulerian Path Johnson 8 Works Cited Biggs, Norman, Edward Keith. Lloyd, Robin James. Wilson, and Leonhard Euler. "Paths." Graph theory Oxford: Oxford UP, This book was written as a textbook for those learning graph theory. The authors did this by including original sources of those influential on the topic. For use in this paper, the focus was on the excerpted source of Euler on the geometry of position. When used for citation in the paper, sometimes it is unclear whether it is from the author's of the text or from Euler himself. However, Euler is addressed in the paragraphs in which his work is cited from this work. Dunham, William. "Hippocrates' Quadtrature of the Lune." Journey through genius: The great theorems of mathematics. New York: Penguin Books, "Eulerian path." Wikipedia. 14 Nov Wikimedia Foundation. 28 Nov < "Graph theory." Wikipedia. 12 Apr Wikimedia Foundation. 28 Nov < O'Connor, J. J., and E. F. Robertson. "Leonhard Euler." Euler biography. Sept University of St. Andrews, Scotland. 28 Nov <

51 Lloyd Shapley and His Work on Game Theory Stephen Kidhardt History of Mathematics Juniata College December 12, 2012

52 Economics is the study of the way societies use and allocate their resources. When being studied and analyzed, the price system is what generally contributes to how these resources are allocated. What happens when prices cannot be the root cause for either legal or ethical reasons, or when prices cause an unstable and crowded market that does not run efficiently and effectively? These are the questions the most recent recipients of the Noble Prize for economics sought to answer. Alvin Roth and Lloyd Shapley produced algorithms and real-life evidence showing a much more effective way to allocate resources in these markets. Their work was in the forefront in market design and stable allocations with the use of cooperative game theory. Game theory has become a branch of economics where mathematical models are used for the study of conflict and cooperation between intelligent and rational people. Many different fields then branched off of game theory including symmetric games, infinitely long games, and differential games just to name a few. Shapley and Roth used cooperative game theory for finding stable allocations in markets. Cooperative game theory assumes that players are able to form binding commitments with each other as well as being able to communicate with each other. John Forbes Nash s work in this subject resulted in an equilibrium which was named after him. This equilibrium states that if each player chooses a strategy and no other player can benefit by changing their solution, while the other player s solutions remain unchanged, then the situation is called a Nash Equilibrium. A simple example of this involves the prisoner s dilemma. Each player has the

53 decision to make whether to cooperate or defect. Each player, however, improves his own situation by switching from Cooperate to Defect. Therefore, the prisoner s dilemma has a Nash Equilibrium where both players choose to defect. This is an important concept in Shapley s work because he tries to find the Nash Equilibrium in given situations. When it comes to skilled labor, allocation can become a problem when trying to match workers with different firms. Since no two workers have exactly the same characteristics, the matching of these workers can become quite difficult. Lloyd Shapley and David Gale sought to find an answer to this problem by creating stable allocations between the two parties. In cooperative game theory a stable allocation is a situation such that no coalition can deviate and make its members better off. This is an important concept in economics because it creates a frictionless marketplace. The theory of stability can then be shown mathematically. Let xⁱ represent i s individual payoff. If the members of coalition S can use their own resources to make themselves better off, then coalition S can be improved on by x or block x. When resources are transferable, coalition S can be improved upon if Σxⁱ<v(s) (1) where v(s) is the coalition s worth. The worth is an economic sum of money that coalition S can generate using its own resources. Then if this inequality holds, S can produce a sum of

54 money and distribute it to make its members better off than they would be under x. Therefore the allocation is unstable. It would then be stable if it was unable to be improved upon. Then the payoff vector would be stable if Σxⁱ v(s) (2) Then the set of all stable payoffs is called the core. However, there can be instances when there is no stable way to allocate the resources. Bondareva and Shapley each derived a formula for how much surplus must be available in order to find a stable matching. The idea of stability leads into Shapley s most well known algorithm, the Gale-Shapley algorithm. Gale and Shapley s work examined two-sided matching. This is a topic in game theory where two disjointed sets of agents must be matched together in order to carry out transactions such as workers and firms, buyers and sellers, or students and schools. An example they examined was marriage. The first example is to assume four women and four men are willing to marry one of the other four, and they rank the other people from the other sex from 1(most desirable) to 4(least desirable). In this example, there can be no ties. A matrix of this could look like

55 Men/Women A B C D a 1,3 2,3 3,2 4,3 b 1,4 4,1 3,3 2,2 c 2,2 1,4 3,4 4,1 d 4,1 2,2 3,1 1,4 where the first number in the pair gives the ranking of women by the men, and the second number is the ranking of the men by the women. Thus, a ranks A first, B second, C third, and D fourth. The goal of their study was to find a way to group one man with one woman in order to create stability. In this specific example, there is only one stable pairing. The question that arose was will there always be a stable set of marriages? The question and proof then became known as the stable marriage problem or the Gale-Shapley algorithm. They started off the proof by letting each boy propose to his favorite girl. Each girl who received more than one proposal would reject all but her favorite from among them. The girl, however, does not accept him, but keeps him on a string and allows for someone she likes better to propose. This is the end of stage one. The boys who were rejected in the first round then propose to their second choice. Each girl who receives more than one proposal keeps her favorite one and rejects the rest. We keep going through this until each person is paired up with exactly one. Since every boy can only propose to the same girl

56 once, every girl is sure to get a proposal in due time. Once this is done, every girl is required to accept the boy on her string. We then know that this is a stable matching, because if boy a likes another girl more than his current wife, he has already proposed to her and has been turned down. Therefore, no one can better their situation by marrying someone else. This algorithm will also produce a stable arrangement in at most n²-2n+2 where n is the number of people in the two-side matching. An example of this kind of deferred-acceptance procedure could be used in college admissions. For simplification, a student that would not be accepted under any circumstances would be automatically denied and would not be considered in this algorithm. To begin, all students apply to the college of their first choice. The college then fills up its quota, q, and puts these students on a waiting list. The school then rejects the rest of the applicants. The students who had their applications denied then apply to their second choice. This process repeats until every student is matched up with a college, thus creating stability. The proof that this is stable is analogous to the proof given for the marriage problem. This matching would be most favorable to the applicants since they get to make their top choices. The symmetric applicant-proposing version of the algorithm leads towards an applicant-optimal stable matching. This illustrates the applicants interests as opposed to the employer. Because of this, applicants agree this is the best way to do things, while employers think this is the worst. The opposite of this would then be

57 employer-optimal stable matching, where the algorithm is sorted to the needs of the employer and not the student. However, in this case, the applicant-optimal matching would be best since schools are supposed to be resources for their students. In the 1980 s Alvin Roth built upon the work that Gale and Shapley did in the 1960 s. The key contribution that Roth added dealt with the evolution of the market for new doctors in the U.S. and that a stable algorithm improves the function of the market. Students who graduate from medical school are often hired as residents at hospitals. Due to the high competition for these residents, hospitals were makings offers to them increasingly early, even sometimes years before a student would graduate. These matches were made before students could produce evidence on how qualified they would be, or what kind of medicine they planned to practice. The market also suffered from congestion; when a student rejected an offer, it was often too late for him to apply for another one. Roth found that this kind of problem was plaguing many markets including the market for psychology internships, dental residencies, and the markets for Japanese university graduates. To deal with the problem in the United States, a clearinghouse was introduced that matched doctors with hospitals using an algorithm which was found to be essentially equivalent to Gale and Shapley s. This program is still in place today and uses a form of the algorithm produced by Gale and Shapley in 1962.

58 Alvin Roth also looked at school admissions in New York City. Prior to 2003, applicants to New York City public high schools ranked the five schools they preferred most and these preference lists were then sent to the schools. The schools then decided which students to admit and reject. This went on for two more rounds with the remaining students entering into an administrative process. This process was very congested with about 30,000 students ending up in the administrative process every year. Another problem with this system was that students were most likely to be admitted if they ranked the school as number one. Therefore, if a student did not have a realistic shot at being accepted into their top choice, it would not be in their best interest to put it as number one. In 2003, Roth helped refine this process. In this new process, it is optimal for the students to report their preferences truthfully, and because of this congestion was eliminated. During its first year in practice, only about 3,000 students had to be matched with schools for which they had not expressed a preference, a 90% reduction from previous years. This helped create a stable outcome and helped reach a Nash Equilibrium. Other schools in the United States also adopted similar algorithms, most recently in the Denver public school system. Lloyd Shapley and Alvin Roth s work also contributed much more to the field of game theory, helping them win the Nobel Prize. Shapley introduced the main single-valued solution concept for coalitional games with transferable utility, now called the Shapley

59 value. This value played a major role in the development of cooperative game theory, which has a large variety of applications, including government taxations and redistributions of utilities. Outside of cooperative game theory, Shapley s research includes mathematics, atomic games, non-cooperative games, and convex games, just to name a few. Alvin Roth continued to take many of the theoretical concepts of cooperative game theory and apply them to real life situations and models. In 1991 he described how laboratory experiments and field observations can interact with game theory, establishing economics as more of an empirical science. Roth and his co-authors tested the prediction of cooperative bargaining theory. Laboratory experiments led him to reveal that subjects change their behavior over time. This led him to create a reinforcement learning model that can predict behavior ex ante. On the surface, the Gale-Shapley algorithm actually involves very little mathematics. One does not need to know calculus, geometry, or even algebra to understand the flow of the arguments and examples of the algorithm. However, the mathematics comes from the flow of the argument and being able to think through the process that the algorithm creates. Mathematics needs not be concerned with figures, whether numerical or geometrical. Mathematics is a way of thought and the flow of an argument. The Gale- Shapley algorithm uses mathematics and in this way, they were able to use their mathematical backgrounds to expand into the world of economics and, with the help of

60 Alvin Roth, work to solve real world problems. Because of this, Lloyd Shapley and Alvin Roth were more than qualified to win the 2012 Nobel Price for economics.

61 Works Cited Shapley, L. S., and D. Gale. "Admissions and the Stability of Marriage." The American Mathematical Monthly 69.1 (1962): Print. Shapley, L. S. A VALUE FOR N-PERSON GAMES. Ft. Belvoir: Defense Technical Information Center, Print. "Stable Allocations and the Practice of Market Design." (n.d.): n. pag. Rpt. in Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences. Stockholm: Royal Swedish Academy of Science, 2012.

62 The Four Color Problem Coloring Cartography Melissa Stanton 12/13/12 History of Math The four color problem asks, [c]an every map be coloured with at most four colours in such a way that neighbouring countries are coloured differently? (Wilson, 2002, 2). It was observed in the 19 th century that one could color in a map of England with only four colors, and subsequently every other map found could be colored in with four colors in this way. We discuss maps drawn on a plane or a sphere (which can be projected onto a plane) so that the maps could theoretically exist with regions in the world. We define neighboring regions to share a common boundary (other than a single point). Therefore, a map such as Figure 1 below is able to be colored in two colors rather than four since the regions only meet at a point rather than a boundary (Wilson, 2002). Figure 1 The four color problem was first posed by Francis Guthrie in 1852 and is sometimes referred to as Guthrie s problem, (Wilson, 2002; Weisstein). Francis showed his find to his brother Frederick along with a proof. Frederick was studying mathematics at University College, London, and showed the theorem and the proof to his professor Augustus De Morgan. Although De Morgan was not satisfied with the proof, he was intrigued by the theorem and wrote to his friend Sir William Rowan Hamilton about it. De Morgan hoped that Hamilton would become interested in the problem and work toward a proof of the theorem, but Hamilton did not show as much 1

63 interest as De Morgan hoped. However, De Morgan was not discouraged, and he continued to write to other mathematical friends and colleagues, attempting to spark their interest (Wilson, 2002). De Morgan explained the intricacies of the problem in his letter informing Hamilton of the theorem. He wrote: Now, it does not seem that drawing three compartments with common boundary A B C two and two you cannot make a fourth take boundary from all, except inclosing one But it is tricky work and I am not sure of all convolutions What do you say? And has it, if true been noticed? (Wilson, 2002, 23-24). This claim by De Morgan can be more easily understood through the diagram below shown in Figure 2. This figure proves that at least four colors may be necessary in Figure 2 coloring a map. Additionally, we can see that there cannot be another country added to the diagram which borders each other country. The blue region is buried within the red, green, and yellow regions, so if we add another country there is no way for it to border the blue region. De Morgan worked to prove this fact, but when he failed to find a proof, he decided to declare it an axiom that if a map contains four regions, each adjoining the other three, then one of them must be completely enclosed by the 2

64 others (Wilson, 2002, 24). He wrote regarding this view to William Whewell, who was a respected philosopher at Trinity College, Cambridge. Whewell went on to publish the first known print appearance of the four color problem in 1860 (Wilson, 2002, 25). It was claimed, at this time, that the need for only four colors was well-known to cartographers, although this may be fallacious since most maps use more than four colors in their colorings (Wilson, 2002). The saga of the four color problem requires Euler s polyhedron formula, which states that [f]or any polyhedron, (number of faces)+(number of vertices)=(number of edges)+2 or, equivalently, (number of faces)-(number of edges)+(number of vertices)=2, (Wilson, 2002,45). Wilson rewrites this Euler s formula using F for the number of faces, V for the number of vertices, and E for the number of edges (2002). Therefore, Euler s formula tells us F-E+V=2. This formula applies to polyhedra, but we can see that we may project polyhedra onto a plane. If we project a polyhedron onto a plane, we find ourselves with one of the original faces as an exterior region in our planar representation (Wilson, 2002). We can use Euler s polyhedron formula for maps if we consider a map to be a projected polyhedron. Therefore, [i]f we include the exterior region, then (no. of countries)-(no. of boundary lines) + (no. of meeting points) =2, (Wilson, 2002, 50). Using Euler s formula, we can prove that [e]very map has at least one country with five or fewer neighbours, (Wilson, 2002, 53). This merely means that there exists at least one 2-sided (digon), 3-sided (triangle), 4-sided (square), or 5-sided (pentagon) region included in the map. Therefore, at least one of the shapes in Figure 3 is included in the map (Wilson, 2002). 3

65 Figure 3 In 1878 Arthur Cayley used this fact in his paper on the subject of the four color theorem (Weisstein; Wilson, 2002). Cayley noted that to go from a map of n regions colored with four colors, the map may need to be recolored in order to accommodate n+1 colors. He used the technique of coloring a map of n+1 regions and then using those colorings but taking out a region, so the map is colored for n regions. We can do this is multiple ways. If we create a patch that covers the intersecting point of more than three regions, we can convert any map into a cubic map so that any intersection point is between three regions. After a coloring is created with only 4 colors, we may shrink the patch back into a point. This patch technique can be shown in Figure 4 (Wilson, 2002). Figure 4 We can also impose a restriction on the coloring so that there are only three colors on the exterior of the map. We can do this because we can always add a region in a ring around the original map. This is shown in Figure 5 (Wilson, 2002). 4 Figure 5

66 These two methods allowed Cayley to begin an inductive argument that if a map of n regions can be colored with four colors, then it may be proved that a map of n+1 regions can also be colored with four colors. We already know that maps may need as many as four colors since Figure 2 proves that some arrangements of regions cannot be completed with fewer than four colors. What is more difficult to prove, is that any map of n regions colored with at most four colors can be expanded to a map of n+1 regions that also has a coloring of at most four colors. Some maps can be expanded easily, so no recoloring is necessary, as in Figure 6, whereas others must be recolored as in Figure 7 (Wilson, 2002). As we can see, the figure with n=6 regions in Figure 6 has a seventh region added (shown in grey) and the additional region can be easily Figure 6 colored with the other regions keeping their original colorings. However, we see an opposing example in Figure 7. This that shows when the original map with n=8 regions has an additional region added (again, shown in grey), some recolorings may be necessary. In simple cases such that shown in Figure 7, the colorings may be easy to fix, but in Figure 7 more complicated maps we need a method to prove that the map with n+1 regions can also be colored with four colors, if the original map is recolored. Here in lies the main issue of the proof of the four color theorem (Wilson, 2002). We can also examine a proof of the four color theorem by contradiction. Therefore we assume that the four color theorem is false and that there exists a planar 5

67 map that requires at least five colors. There must be a map with the smallest number of countries that cannot be colored with four colors, but can be colored with five or more. We can call this map a minimal criminal, (Wilson, 2002, 68). We can easily prove that a minimal criminal cannot contain a digon or triangle as shown in Figures 8 and 9.These figures demonstrate that since there are four colors, and since there will always be a color available, we can create a map coloring that requires no more than four colors for maps with a digon or triangle. Therefore we have proven that neither of these shapes can be a minimal criminal (Wilson, 2002). Figure 8 Figure 9 This argument does not hold for maps with a square or pentagon. If we attempt to apply the same logic, we see that it is impossible to prove. This is demonstrated in Figures 10 and 11 (Wilson, 2002). 6 Figure 10

68 Figure 11 Since the maps with squares and pentagons have more than four regions, we may find ourselves with no spare color to use when the interior region is added back in (Wilson, 2002). Therefore, we cannot easily disprove that some map needs more than four colors (Wilson, 2002). Although we cannot prove that every map needs up to four colors to color it, we can easily prove that there is no map that requires more than six colors. We know that any map has at least one country that borders five or fewer regions. We also know that any map has a minimal criminal. In this case that minimal criminal can not be colored in six colors but any map with fewer regions can be colored with six colors. Based on the method displayed in Figure 11, we can disprove the possibility of this occurring. If Figure 11 shows our minimal criminal, it shows a map of n regions that needs at least six colors but if there were in n-1 regions, it could be colored in fewer colors. As we can see, we can reduce the number of regions by one and color the map in with at most five colors. However, we notice that this is not a minimal criminal because when we add the last region back into the map, there is an available color for it (in this case, grey). Therefore, we have proven that every map can be colored with six or more colors (Wilson, 2002). The first proof since the problem was presented by Guthrie in 1852, comes from Alfred Bray Kempe in 1879 (Weisstein). Kempe studied under Cayley at Trinity 7

69 College and graduated in Cayley interested him in the four color theorem, and in 1879 he described a method for coloring any map with four or fewer colors. He provided the following steps: 1. Locate a region with at most five regions bordering it. 2. Cover this country with a patch. 3. Extend all the boundary lines so they meet at a point over this patched region. (Thus the map of n regions now has n-1 regions.) 4. Repeat this procedure with the new map of n-1 regions. Continue repeating the process until there is only one region left in the map. Thus, we have reduced a map of n regions into a map of 1 region. 5. Color the 1 region with any of four colors. 6. Reverse the process until the map of n regions is restored and is fully colored. Whenever a new region is added in, color it with any available color. This final step has the same issue as we demonstrated before there is no easy way to prove that a map colored with at most four colors can have a region added and still be able to be colored with at most four colors. This is where Kempe introduced Kempe chains. As we know, if our restored region has at most 3 boundary lines, we can easily color the restored region (as shown in Figures 8 and 9). Again we see an issue with regions with four or five boundary lines (Wilson, 2002). Kempe introduced the method of Kempe chains, which looks at the central region and finds two regions (in our example red and green) that surround it and are not adjacent. We then look at chains (or perhaps branches) of red and green colored regions stemming from those surrounding the central region. There are two possible cases for the chains: the two red-green chains are separate from one another, or the 8

70 two chains link up. In the first case, we have a situation such as that in the first diagram of Figure 12 (Wilson, 2002). In Case 1, the red-green chains above the central square (in grey) do not meet up with the red-green chains below the central Figure 12 square. Therefore, we can alter the ordering of red and green regions above the central square and create a complete coloring. We find that our central square may be colored red. In Case 2, the red-green chains above the central square do meet up with the red-green chains below the central square as shown in Figure 13 Figure 13. Therefore, we cannot alter the coloring of the red-green chain to find a color for the central square. We now turn our attention to the blue-yellow chains. We know that the blue-yellow chain to the right of the central square cannot meet up with a blue-yellow chain from the left of the central square because it is blocked by the closed red-green chain. Therefore we can switch the yellow and blue colorings in the yellowblue chain to the right of the central square and then we find that we can color the central square blue (Wilson, 2002). 9

71 Kempe attempted to apply this same logic to a pentagonal central shape, but there was a flaw in his logic, and therefore, a flaw in his proof. He begins by claiming that if you have a pentagonal central region, in which a chain (in the case of Figure 14, we discuss the red-yellow chain) above the central region does not meet up with the chain of the same colors below the central region, that you may switch the colors of the Figure 14 chain above the central region and color in the central region with the available color. If however, the red-yellow chains above and below the central region do meet up, then we must refocus our attention on the red-green chain. As we can see in Figure 15, we cannot alter the redyellow chain, but we can alter the red-green chain. If the red-green chains above and below the central region do not meet up, Figure 15 then we may switch the colors of the chain above the central region and color in the central region with the available color. If the red-green chain above and below the central region do meet up as in Figures 16 and 17, then we must alter our focus once again (Wilson, 2002). 10

72 We must turn to the blueyellow chains. We know the chain to the right of the central region cannot connect Figure 16 to the blue-yellow chain on the left of the central region, because it is blocked by the red-green chain. Therefore, we can switch the colors of the blue-yellow chain to the right of the central pentagon (as in the second diagram in Figure 16). We still do not have an available color for the central region, so we must examine Figure 17 the blue-green chain. By the same logic as before, we know that the blue-green chain to the left of the central region cannot meet up with the blue-green chain to the right of the central region and we can switch the colors in the chain (as in the first diagram in Figure 17). Now the central pentagon borders only three colors: yellow, green, and red. We finally have an available color for the central region and we color it blue (shown in the second diagram in Figure 17). Kempe believed that he had proven all possible cases and had proven the four color theorem (Wilson, 2002). In fact, his proof was considered valid for eleven years until Percy Heawood found an error (Weisstein; Wilson, 2002). Heawood did not propose a proof of his own; he merely brought the error to the attention of the mathematical community. He showed a counter-example to Kempe s technique using a map similar to that in Figure 18. The counter-example we will examine has twenty-five although Heawood s original counter-example used 18 11

73 countries, and the fallacy can be shown on a map of only 9 regions (Weisstein, Wilson, 2002). While it is possible to color these maps with four colors, they cannot be colored according to Kempe s method. Therefore Heawood disproved Kempe s proof without disproving the theorem itself (Wilson, 2002). Figure 18 12

74 Kempe s method would have us recolor the five regions surrounding the central region so as to have a color available for the grey central region. We may switch the colors of the red-green chain above the central region, as shown in Figure 19, since it does not meet the red-green chain below the central region. We may also switch Figure 19 Figure 20 the colors of the red-yellow chain below the central region, as shown in Figure 20, since it does not meet up with the chain above the central region. Although neither of these color switches causes issues for the map coloring on its own, Kempe s method would allow both of these colorswitches to occur simultaneously. When we examine the map that is a result of applying both color switches at the same time, as in Figure 21, we find ourselves with two red regions touching on the right side of the map. Therefore, Heawood disproved Kempe s proof with a counter-example as we have shown (Wilson, 2002). 13

75 Figure 21 Heawood went on to prove the five color theorem. Although it is not as strong as the four color theorem would be, it was a good result, and was stronger than the previous six color theorem. As expected, the five color theorem states, [e]very map can be coloured with at most five colours in such a way that neighbouring countries are coloured differently, (Wilson, 2002, 125).We can use contradiction to prove that there are at most five colors necessary to color any map. We first assume that there is a minimal criminal of n regions that needs more than five colors, but if we remove one region, any map of n-1 regions can use only five colors. As we have stated before, we know that every map has at least one region that borders five or fewer regions. We know that if the minimal criminal has a digon, triangle, or square, we know that we can delete a region and color the n-1 remaining regions easily. When we add the region back in there will be an available color for it since there are five colors total. This argument is more complex if the map includes a pentagon (Wilson, 2002). To solve the case of a pentagon in the minimal criminal, we again look to Kempe chains. As in Kempe s proof, there are two cases. In Case 1, the red-green chains 14

76 above and below the central region do not meet up and we can switch the colors of the red-green chain above the central region and have a color available for the central region. This can be seen in Figure 22. If however, the red-green chains do meet, then we must turn our attention to a different Figure 22 Kempe chain in Case 2 as demonstrated in Figure 23. We turn our attention to the blue-yellow chains to the left and right of the central region. We know that if the red-green chain meets, then the blue-yellow chain cannot meet. Therefore, we can switch the colors of the chain to the right of the central region and we have a color Figure 23 available for the central region. Therefore, Heawood proved the five color theorem using a Kempe-inspired method (Wilson, 2002). As we have already discussed, there must be a digon, triangle, square, or pentagon (depicted in Figure 3) existing in every map, and the minimal criminal cannot contain a digon, triangle (as show in in Figures 8 and 9), or square (as explained in Kempe s proof and depicted in Figures 12 and 13). Therefore, a minimal criminal must contain a pentagon. Euler proved that every cubic map that contains no digons, triangles, or squares must contain at least twelve pentagons, (Wilson, 15