Sunday, December 6, 2015

Checkers is Solved

I found an interesting write-up about the familiar game of checkers that has an unusual mathematical result. Like Tic-Tac-Toe, when played perfectly, it is a no-win game. This means that if both players make no wrong moves, the game ends in a draw. After all my years in grade school playing checkers on break, I never had a game end in a draw, unless the teacher called an end to break or one player got mad and pushed the board off the table. So what is a draw in checkers? To my imagination, it would be when each player has one piece left, so that one player could endlessly run away from the other as to not get jumped, or something to that extent. Called a tie, it was thought to be impossible so make the game end in a draw. 

Checkers is most like a military battle. The strategy, written by hardened veterans, deals with the analytical side of checkers. Being able to quickly calculate the possible moves at hand gives an advantage to the player. Being able to calculate all possible seems to be the common trend among experts and they calculate this quickly with an mXn rule; the number yoyur checkers left on the board times all possible moves each one can make. But, there is the opponent element. You could say that your opponent will probably do this, or probably do that, but you can never know until he/she does move. That is why being able to invision the next move, and the next move, and the next, proves to be an advantage in Checkers.

In 2007, Jonathon Schaeffer and colleagues used computers to prove the result. It took hundreds of computers and 18 years to prove the most complex game ever solved. A solved game is a game whose outcome (win, lose, or draw) can be correctly predicted from any position, given that both players play perfectly. Games which have not been solved are said to be "unsolved". This is a mighty feat considering there are roughly 5 x 10^20 possible positions or arrangements. How did they do it? The team considered 39,000 billion arrangements with 10 or fewer pieces left on the board and then determined if red or black won. Then, they used a special search algorithm to study the start of each game and to see how moves funneled into the 10 checker configurations. The solving of checkers represented a major benchmark in the field of artificial intelligence. 

So what did they do with this research? Can you think of anything better than matching the computer up against world champion checkers players? Nope! The program they called Chinook played former world champ Marion Tinsley to a series of draws and other respected players consisting of 40 matches. Chinook has a website and you can actually play him/her. Be prepared to lose... at best, tie.

Checkers was not the first board game to be "solved." Many others including connect-4, tic tac toe, and awari have all be solved. There is also some information on the webpage detailing the specifics of the programming. Proof number search is best-first search algorithm used to solve many games. What!? Even a link to solve your own game. Solved games and helping you solve yours

Sunday, November 15, 2015

Euler's Number e

In conclusion, the discovery of this mysterious number certainly raised more questions than it answered. After Euler's discovery, e started popping out everywhere and over the next few hundred years became the foundation for several applications in probability. We see the biggest impact in the finance industry. Brokers use algorithims involving e to make their clients rich, banks use it to determine your interest, and insurance companies use it to set your premiums. All in all, since the world revolves around money and one's ability to acquire it, e is basically big brother of the root of all evil.

Sunday, October 18, 2015

A Short Journey through Genius Summary

Journey through Genius: The Great Theorems of Mathematics.

 I'll say this book became was an interesting read for a guy that hasn't read a book for pleasure in years. The history, mathematical methods, and sheer intelligence relayed in this book made for a well rounded, exciting read. This book takes us through the start of mathematics, although it jumped quickly through everything up until Hippocrates. It breaks down and illustrates the evolution of what it means to do math, both practically and theoretically. And, gives us insight into the greatest minds of history. From pure will and concentration of Isaac Newton, to natural talent and common sense of Euler. From public debates and "math offs," to winner take all published math problems, this book had it all.

We start off with a history of mathematics. From early number systems and counting techniques, to mentioning some of the great mathematicians before Hippocrates. The first chapter sets the tone for the book and already we can see some of the great strides in mathematics.  Next, Plato creates enthusiasm for mathematics, making it a noble field of study. Exodus who was known for theory of proportion and proof by exhaustion, saw a practical use for mathematics in the branch of astronomy and planetary motion. Then, Euclid, the most notable up to this point, gathered and perfected all know mathematical conjectures with his Book of Elements. In a very precise, non-circular way, he proved 465 theorems, one after the other, most relying on the proof that came before, with just 25 definitions, 5 postulates, and 5 axioms. Arguably the most studied book for centuries to come. All 465 theorems still hold true to this day. Worth noting in this summary is Postulate 5 concerning parallel lines. Now we realize how initiative and clever these guys were. In Chapter 3 we get a inside look at Euclid's book of elements: Euclid set the bar for geometric algebra, plane geometry, and number theory. In book 8 we see a great theorem in the form of number theory concerning prime numbers and their infinitude. In Chapter 4 we meet the great engineer and absent minded Archimedes who studied Euclid. He developed a genius way to prove the area of a circle. He was famous for shrinking the number of cases in a proof and favored the proof by contradiction. He united all circular shapes the constant pi and estimated it to within 2 decimal places and with his method and computers we now know pi to half of a billion decimal places. By chapter 5 we are introduced to Heron and his great formula for calculating triangular area. Here is the point where things started to click for me. I started putting the pieces together from what I had learned early in mathematics to what I know now about what it means to be a mathematician. Then, in chapter 6, we learn how doomed Cardano was from the start, yet gives us the formula for solving cubics. A trend starts to develop in the form of diversity. From many different backgrounds and nationalities, these mathematicians are as diverse as the fields in which they study. From Geometry, to number theory, and from application to theoretical; the branches of mathematics continue to grow and spread out as each mastermind finds his niche. Newton in physics and then the Bernoullli brothers in analysis, The 1500-1700s saw an explosion of great minds. Some great minds studying under the previous, some enemies and some with a competitive spirit. Euler is now my favorite mathematician after reading this book. He overcame some physical set-backs to contribute half of the know theorems in analysis, and up to 1/3 of all published text up to this point all the while remaining a good guy who cared about education and put forth the effort to convey his thoughts and practices in an easily understood manner for any pupil.

Overall, this book has both inspired me and intimidated me. Oh how I wish I could have studied under Euler, heck, I'd even take Newton. I'm a believer in a growth mindset and that with enough time, anyone can learn anything, however the concentration and determination of the mathematicians outlined in this book is not a skill that can be learned, just admired. After reading this book, I have a greater appreciation for old fashioned, pen and paper mathematics. Not only did this book provide us with the history of math and those who practiced it, but told a story of perseverance, and patience. I would recommend this book to anyone who decides to major in mathematics.

Monday, September 28, 2015

Around the Bases

I was bothered by a prompt concerning our number system. Why is this base 10, with 9 distinct symbols, system of counting the one in practice today? After some initial research, I found many other systems that seem just good. What could make one system better than another? How would the nature of mathematics have changed given a different system in place? Let's go through some of the history and make sense of how and why we count to ten.

The Babylonians 3100 BC: A base 60 system? seems absurd right? 
The Babylonian numeral system is credited as being the first positional numeral system. This means the symbol, and its position determined the value. A milestone in human history. We still see the effect this new development on time. 60 seconds in a minute, 60 minutes in an hour, however not exactly the same. Since arithmetic for large number, and powers of numbers in this system were not easy, it is no wonder it didn't survive. I look at this and see that calculations would be similar to that of Z[60].

The Egyptian Numerals 3000 BC: aha, a base 10 system.
Value1101001,00010,000100,000million, or


The difference here being that position didnt matter. However, surprisingly, by 1740BCE, the egyptians had developed a symbol for zero. They thought of this as a baseline and eventually thought of numerals as either positive and negative. Also, they even had symbols for fractions,  a huge contribution to how our base 10 numeral system works today.

The Maya numerals: 

A base twenty system in which 5 dots make a bar, and numbers after 19 were written vertically as powers of twenty like so:

So, their place holders were of vertical orientation. The use of bars and dots made arithmetic very straight forward. The Maya numeral system is most famous for its appearance in the Long Count Calendar. Our numeral system today doesn't reflect much of genius system the Maya used, however their use of zero rivaled that of their predecessors. 

The Roman Numerals 1000BC: Probably the most well known historic system of counting since some small applications using the roman numerals still exist today.
Surprisingly, this system did not have a 0. So the idea of negative numbers did not exist. Further, instead of writing out 4 I's, like IIII, the Romans used a sort of subtraction like IV, which we would say 5 minus 1 today. Aside from still using the Roman Numerals for basic applications like numbering lists, or telling time, the system was outclassed by another for it's ease of arithmetic and basic operations.

In a sense, the most widely used numbering system today, the Arabic numeral system, was not invented overnight. The Babylonians told us position matters. 61 is certainly different than 16 in our system. The Egyptians had a base ten for ease of application into multiplication and division, ours is also a base ten. The Maya had a base twenty, but knew that higher mathematics required very large numbers, something the Arabic systems also handles well. And the Roman Numerals, well... they are a special breed, much like the Chinese counting rods. These are used as special, or ceremonial techniques. This may be because they look nice, and can be detailed to show such special meaning.

It's was Fibonacci's famous book, Liber Abaci, that changed the way people described quantitative elements. As Fibonacci traveled the Mediterranean, he fell in love with a number system popular with Hindu and Arabic merchants.  The Hindu Arabic numeral system was born and Fibonacci set out to make it known to the world. In his book, he provided detailed instructions on how to convert from another system to the HA system, along with providing details on how to use this system in everyday applications like weight, money, and other quantities. He followed up with a second section on business, which helped economies grow. 

The role of our number system today: One thing I notice different about our modern day system is the decimal system. Of course, with a lot of work, any of the above systems could make numbers as big as they like,  but our system can handle values as small as we like. The role of our number system is especially important in application of very small numbers; like in chemistry or physics. In my opinion, the system we have today opened the door for the exploration into the hard sciences which in turn lead to a more advanced, technological civilization. However, scientist and researchers are still coming up with better number systems as the development of computer science field progresses. Traditionally, a binary system, 0s and 1s, is used in computer programming. However, some claim other systems, like that with base 16, can provide better programming and data storage. Who knows, maybe with the next 25 years we will be counting to VI (16) instead of 10.

Sunday, September 13, 2015

History of Mathematics

The Greeks and Proof Writing.

Throughout my mathematical career and my experiences with proof writing, I'm convinced the Greek mathematician like Thales, Euclid, and Pythagorean made the one of the biggest contributions to communicating in mathematics. Before the Greeks, civilizations like the Egyptians and Babylonians relied on repeated observations which is what we now call inductive reasoning. The Greeks however, not only gave us geometrical theorems, but also dabbed in number theory as well as formalizing the proof process.

First, a conjecture is proposed and any mathematician will admit to trying lots and lots of examples. Sometimes, these examples will lay the groundwork for how the proof will flow. Other times, the actual proof is so abstract that even the simplest of examples can be troublesome to construct. Nevertheless, if the conjecture or proposition is proved using arbitrary elements of the given set following the axioms, then it becomes a theorem. Theorems laid by the Greeks are still in practice today. In my mind, theorems are like shortcuts; once someone has proved that the theorem holds, we can skips all steps in between and cut to the chase if we are given a problem with the same hypothesis. For example, a theorem may start out with "suppose we have a right triangle, then..."; already this grants us many avenues in which to work with the triangle since there are many theorems pertaining to right triangles.

Over the years, proof writing has become an art form. This may be because of Thales and his proof by picture. With explanation, a picture is a powerful tool. However, this limits one to thinking abstractly. Maybe that was his way of convincing his counterparts that he was right. Euclid was famous for Euclid's lemma and other number theory problems but he also made of compilation of the work of famous mathematicians that came before him. This collection was probably the most influential piece of literature for other mathematicians that came after Euclid.

All in all, the Greeks contribution to proof writing has evolved mathematics into a more diverse field.

Wednesday, September 2, 2015

What is Math?

What is Math?

When the general discussion of Mathematics gets started, which is always the case when peers ask me what my major is, the result quite often follows the same routine: "Oh, so you must be really good at multiplying and adding?" No, I am not a calculator. This usually goes on for several minutes before I mention studies in Abstract Algebra like Group theory or Ring theory. Math is much more than just numbers, it can be used theoretically to answer some of worlds most unexplainable phenomenon. We are in the age of information where researchers and engineers are making breakthroughs everyday using advanced computes powered by mathematical formulas and theories. From the first recipes using quantitative measurements, to the millions of lines of code in a program to predict weather, math has evolved to shape civilization and the way we see the world around us.

Mathematics can be seen everywhere in everyday life, even back to the stone age. For example, humans were probably using algebra before it had an official name. How many deer must we kill in order to feed the clan of 100 members for one week when each person eats one-eighth of a deer a week. The advancement of Algebra made it possible to predict, plan, and concur the extreme living conditions of early man. On the other hand, some of mans greatest accomplishments were possible because of Geometry. Pyramids can be found all over the world and the great pyramids in Egypt required more geometry and algebra than meets the eye. Without Calculus, Physics and advanced Statistics would not be possible. Each milestone is similar to a building block. Down the timeline, each block is built upon another supporting the overall structure. Like our human civilization, mathematics will continue to grow and evolve.