A Magical Birthday Square (Doing Math

After learning about the great accomplishments Srinivasa Ramanujan did with such limited resources and at such a young age I was astonished. Then I found out he came up with a magic square with 22, 12, 18, and 89 in the first row, which is his birthday! I thought it was so cool that I had to try it for my own birthday. So my first row consisted of 1, 7, 19, and 92. That meant my target sum was 119. I wasn’t sure where to start at first and thought I was going to have to do a lot of guessing, but then I saw that when finding the target sum of the top right corner of the 2×2 I had 19 and 92 already. When you added those two up you get 111, which meant the two bottom squares of the 2×2 could either be 6 and 2 or 5 and 3. This made guessing a lot easier, so I went with 6 and 2, with the 6 under the 19 and 92 under 2. I did it in this specific order because I will need those rows to add up to 119 as well and since 92 is such a big number I wanted to place it with as many small numbers as possible. (If at any time you get confused on where I put any numbers, there is a picture of my square).

                After I made this guess the rest of the 2nd row was easy to figure out because I just had to look at the top middle 2×2 and the top left 2×2. So for my second row I had 24, 87, 6, and 2. This adds up to 119. I then looked at the diagonal that had a positive slope. It already had 92 and 6 so the sum was 98. This meant that my possible options were 3 and 18, 4 and 17, 5 and 16, 9 and 12, or 10 and 11. Not quite as nice of odds for guessing so this is when I looked at the four corners. I already had 1 and 92, which is a sum of 93. I knew that of the numbers in that could be in the diagonal had one of the same number as the four corners. So the possible numbers for the bottom to corners was 3 and 23, 4 and 22, 5 and 21, 8 and 18, 9 and 17, 10 and 16, 11 and 15, or 12 and 14. That unfortunately didn’t eliminate any so then I looked at the last column which had 92 and 2, a sum of 94. I was hoping this would eliminate some possible options for the four corners. So the possible options for the last column were 3 and 22, 4 and 21, 5 and 20, 8 and 17, 9 and 16, 10 and 15, 11 and 14, or 12 and 13. At this point I’m frustrated because I am not catching a break, so I go back to the diagonal I looked at originally and I took a guess. I choose 10 and 11 thinking that they were fairly neutral numbers (neutral meaning they won’t screw up the whole square by a lot if the numbers didn’t fit). Now that I had that I could find the middle 2×2 since I already had 87, 6, and 11. So I got 15. This allowed me to find the other diagonal since I had 3 out of the 4 squares and I got 16. I can find the first and last column since I had only one square blank in each so for the first column I had 84 and the last column I had 9. From there I was able to fill in the rest of the square.

                Once I filled all the squares in, I started to check my work. I did this while in my MTH 122 class and as I added up each row getting a total sum of 119 my heart starts to speed up. I then add up each column and again a total sum of 119 for each. My heart I start to get a rush of adrenaline and my leg starts shaking. I think to myself, “Could it be that my birthday, just like the great Ramanujan, makes a perfect square!” I think check my diagonals, again 119! I start to lose my breath. Then I check the middle 2×2 and the middle edges. All add up to 119. At this point I’m freaking out. I have to try and contain my excitement since I’m sitting in class. I then check the top middle 2×2 and bottom middle 2×2. Boom! 119 like a champ! I then take a picture and post it in our Facebook group. After I made the post, I start to double check my work… I see I didn’t check the left middle 2×2 and right middle 2×2. And they don’t look pretty. My numbers are way off. Instantly my dreams are shattered. All the excitement I had been feeling deflates at rapid speeds. I start to look at what I could change to fix the square so it would work and I realize that it is impossible. Since 1 is in the first column I need to have a big number somewhere in that column. Ideally, I would want to put it in the bottom square so I wouldn’t screw up the middle right 2×2, but because 92 is already in the top right corner I can’t do this (because 4 corners and the diagonal wouldn’t do this). This means that I have one really big number for the first column, which is fine I can accommodate for that. It is not ideal but it still can be done, but because the top right corner 2×2 under the 19 and 92 can only be 2 and 6 or 3 and 5 it means that the top middle have to be a large number. Thus you will always have two really big numbers putting you over 119. I unfortunately was unable to get my birthday work, but I came close and got to feel like I was with the elite, like Ramanujan, for at least 5 to 10 minutes. Maybe the birth of my kids will be in the magic square club.

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Georg Cantor the Great

I was very excited to look at the work Georg Cantor did for math in class. This was fueled by several reasons. For instance, Georg Cantor was extremely controversial with his discovery of multiple infinities. Also, I read about him in Concepts of Modern Mathematics by Ian Stewart and I’m doing my final project on the guy. For the sake of time and lengthy explanation needed to explain countable infinities and uncountable infinities I am going to assume that you already have an understanding of these concepts.

So as I previously stated, I am doing my final project on Georg Cantor and his work on infinity. I have looked into the Continuum Hypothesis, which says “There is not a set whose cardinality is strictly between that of the integers and the real numbers”, with the understanding that most mathematicians believe that they hypothesis is true, false, or impossible to determine. It is always fun to have people disagree, which is probably why I took such great interest in Cantor’s work. But in reality when researching and understanding these two separate infinities, and their abstractness altogether, I thought of it as fun, but pointless stuff. I don’t mean to try to take away the weight of the remarkable fact that there is such a thing as countable infinity, but I saw no significant reason in this being a huge discovery.

Well I was wrong in my thinking because after reading Concepts of Modern Mathematics I found that infinite cardinals were the key to proving the existence of transcendental numbers and how they compare to algebraic numbers. (A algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. A transcendental number is just the opposite. So the square root of 2 is an algebraic number, but pi and e are both transcendental numbers.)

Cantor proves this first by taking the height of a polynomial. It should be noted that there is a finite amount of polynomials that can be created with any specific height given. Cantor goes on to show that for an arbitrary polynomial, any whole number m we define f(m) to be the (m+1)-th distinct algebraic number along the sequence. This makes f a function N to the set of algebraic numbers. He proves that f is a bijection, which means the set of algebraic numbers has cardinal aleph-zero. Cantor knew that there were real numbers in the algebraic set such as the square root of 2 and he knew that the real numbers form an uncountable set. So this proved that there were some real numbers that are not algebraic, thus proving the existence of transcendental numbers. It also showed that there are more transcendental numbers than algebraic numbers. Suppose there were aleph-zero transcendental numbers, which would mean that

aleph-zero (algebraic) + aleph-zero (transcendental) = c(cardinal of R)

That is not true because aleph-zero plus aleph-zero equals aleph-zero. This was revolutionary because mathematicians had believed that transcendental numbers as being very rare and this proved that there was significantly many more than algebraic. This showed that almost all real numbers were transcendental. *Mind blown*

I am surprised that I did not come across this with my research on the internet. I found this gold mine by reading a book that just so happen to touch on Georg Cantor’s work. I think that the Continuum Hypothesis has casted a large shadow over his work on proving transcendental and algebraic numbers unfortunately.

Bracket Perfection

“While there is no simple path to success, it sure doesn’t get much easier than filling out a bracket online.” These boarder-line philosophical words were uttered by the great billionaire Warren Buffet. As all Americans have heard, unless you live under a rock, Buffet has offered one billion dollars to whoever makes a perfect bracket for NCAA March Madness tournament. It has surprisingly gotten quite the hype and causing people who would never fill a bracket out giving it a shot.

I myself have participated in March Madness for the last 3 years now and being the mathematician I am I have already done the research for the perfect bracket. It seems every year there are outrageous rewards from people to whoever fills the perfect bracket. Unfortunately without much digging one will find that no one has EVER filled out a perfect bracket. The odds are 1 over 2 to the 63rd power (that’s 1 in 9 quintillion).  With odds like that I would be willing to offer my first born child to whoever gets the perfect bracket. With that being said that does not mean I have not tried to create the perfect bracket… and I think I succeeded.

I used a little history to help make this bracket. For instance, a number one seed has never been beat by a number 16 seed. So, all four of the first seed teams will win round one. That makes picking 4 out of the 63 games easy. That means only 59 more games to pick. Now things get a little tricky; only once in history has a team seeded higher than 11 made it to the elite eight. I think this makes it safe to say that the seeds 12-16 won’t make it to the elite eight; meaning they will lose somewhere before then. This also means in the elite eight seeds 1-11 could be in it. For each region 2 out of the 11 teams will be in the elite eight. There are 230 teams that have been in the elite eight in the last 29 years.

Seed # % to Elite Eight
 1 34.783%
2 23.487%
3 13.043%
4 7.826%
5 3.478%
6 5.652%
7 3.043%
8 3.043%
9 .869%
10 3.043%
11 1.304%

We see that seeds 1 and 2 have the largest probability to getting to the Elite Eight. To be honest though there will never be a time in March madness where all the teams in the elite eight are all 1 and 2 seed teams. This means we have to step away from the math and guess. In each region you need to pick a 1 or 2 seed team for the elite eight. Typically if they are in the big 10 conference you should put them through…well unless it’s Michigan (they typically choke). As for the other team for each region I usually skip around by picking at random.

After you establish your elite eight you must make educated guesses for the games that were in the sweet sixteen and second round. You must expect upsets so don’t pick the lower seed. For the elite eight I have two 1st seeds, two 2nd seeds, 3rd seed, 4th seed, 10th seed, and 8th seed.  The odds of these seeds going to the final four are

Seed # % to final four
1 41.228%
2 21.93%
3 12.281%
4 11.404%
8 3.509%
10 0

Again you’re not going to get all 1 and 2 seeds in the final so you won’t pick all 1’s and 2’s. We can clearly see the 10th seed is out of the picture. I think it would be safe to both 1 seeds, one of the 2 seeds, and the 4th seed. For the final game you want to a number one seed and then random so we will go with 2. For the final you need to flip a coin, a head is seed 1 and a tails is seed 2. In this instance we go 2. So my bracket looks like this

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Now for each region you can you could do the same steps. For instance, this could have been Michigan winning it all. Essentially you must be the best guesser in the world for all the possible options.

Book Review on Concepts of Modern Mathematics by Ian Stewart

I read “Concepts of Modern Mathematics” by Ian Stewart. I was stoked about reading a book on math because I enjoy reading and I enjoy math. Basically I equated it to a simple conjecture; If I like reading and math, then reading math will be sweet! I had read that my boy Ian Stewart was a funny writer and that this book was a good example of that. This was good. It made it look like the conjecture I had come up with would be true. Before I would be sure if my conjecture was true I had to actual read the book. And that is exactly what I did.

            I finished the first chapter enjoying Mr. Stewart’s voice and fun joke. He had a good mix of puffing up mathematicians to being the most superior beings in the world and to how abstract/ludicrous they could be (especially theoretical mathematicians).  After finishing the first chapter I was intrigued by what would come and would show the book off to all my “inferior” friends. I would even occasionally read it in my college algebra class (which I have a B in).  This first chapter confirmed my conjecture. Now I couldn’t make this a theorem until I read this whole book, but I at this point I was a bad mathematician and cut corners and assumed that it was true.

Since I believed that my conjecture was true it made me make a lot of excuses when I ran into things that proved my conjecture false. This false-belief in my conjecture helped me read the next 5 or 6 chapters with much… perseverance. Chapter two, just coming off the high on the first chapter, was interesting to an extent, but I started to have a decreasing slope in my interest in the book. I didn’t want my conjecture to be wrong, but was shaping out to be that way the more I read.

Mr. Stewart slowly became less of my friend who I wanted to hang with and more of a dreaded professor whose class you have to sit through every week. The book covers a very large amount of math. Much of it was review from previous classes, such as advanced algebra (mth 310) and discrete mathematics (mth 345). This typically made me want to fall asleep because I was not interested in relearning information I already knew (which is ironic because I am taking college algebra, but hey I sway in some of my logic). Also, the book hit a lot on the information taught in the two classes I mentioned above and these were two classes I wasn’t overly zealous about. It often had me remembering the suffering and pain I endured taking these classes.

Although I did not enjoy the book and unfortunately my conjecture was wrong, there was some interesting parts of the book. It was like digging for gold. You have to read a lot to find little gold nuggets in the reading that you enjoy (probably not large enough to have any significant values, but it is gold none the less). For instances, I found that his chapter on there being two types of infinity was interesting to me. I was really impressed when he proved that you can count rational numbers , which unfortunately would take a whole page and a half to give the context and explanation to show how he did it. So the book wasn’t all bad. I would say that there around 15-20% of the information that is interesting and captivating, but overall the book was better at putting me to sleep. Though my original conjecture was wrong, I have revised it and I will test on this next book I read this semester.

Conjecture: If I enjoy reading and math, then I may enjoy reading a book on math.

History: Fibonacci

Leonardo Pisano Bigollo, more commonly known as Fibonacci was a great mathematician in the early 13th century. He is most known for his famous number sequence, the Fibonacci sequence. This is a fantastic sequence and surprisingly has a connection with the golden rule. Although Leonardo is most commonly known for this, this is not his most influential and prominent work in mathematics. His most impactful work is used every day by millions of people around the world. He was the head honcho in spreading the Hindu-Arabic numeral system in Europe.

            It is commonly believed that Fibonacci was born to a wealthy Italian merchant. His father directed a trading post in Bulgia. Leonardo would travel with his papa to help. It was during this time traveling with his father that he learned about Hindu-Arabic numeral system. After studying throughout the Mediterranean world, he returned to Europe, bringing with him Hindu-Arabic numeral system. Europe was still using Roman numerals at this time. He used his book, Liber Abaci, which literally means Book of Calculation, to help spread this numeral system.

            Now Fibonacci’s claim to switch from Roman numeral system to Hindu-Arabic numeral system could have easy got him laughed straight out of Europe. He had to market this new system well if he didn’t want to look like a fool. He accomplished that through the use of his book previously mentioned. In his book he showed the simplicity in mathematical computations by applying this numeral system to commercial bookkeeping, conversion of weights and measurements, the calculation of interest, and changing money. Clearly working with his papa as a young boy had a large impact in his life and his knowledge in the market place helped create a strong foundation for this new numeral system to launch off of in Europe.

            The aim of the book was to be able to do calculations without the assistance of an abacus. In laymen terms it was to make math easier and that exactly was Fibonacci did. His book can be broken into four sections. The first section he introduces the Hindu-Arabic numeral system. The second section gives practical examples, such as conversion of money and measurements, and calculations of profits and interest. The third section talks about several mathematical problems. One problem in this section was on the growth of a population of rabbits. This is the origin of the Fibonacci sequence. The fourth section derives approximations of irrational numbers such as square roots.

            This book, I believe, put the study of mathematics into a theoretical incubator. Math from this point starts to progress in its discoveries because it allowed much more ease when doing complex math.

 

            On a side note: 2002 marked the 800th anniversary of Liber Abaci. It was published for the first time in modern English. The book, on Amazon.com, cost $70! Even though it is 800 years old, Liber Abaci was and still is extremely significant to the science of mathematics.

Doing Math- tessellations!!!

So I created a tessellations. I made it myself with my own hands. This was much hard than expected, i think it was due to my perfection expecting. To make the tessellation I created stencil using a note card. I drew a pattern on the note card and then used a box cutter to create space to draw through my pattern. Well the blade was to thin for my pencil/pen to fit through so i ended up putting my paper and stencil  on a note book creating a soft back. I then preceded to jam my pen on the stencil draw over the pattern over and over. This created an indent on my paper so i was able to then trace the indents with a pencil. I eventually started to ruin my stencil and it fell apart. This is when i had to stop my tessellations. I found that when i tessellate i tessellate hard.

photo 1photo 2 photo

The Golden Ratio

The golden ratio is 1:1.618. We see that 1.618 is often represented as phi. The origin of the golden ratio is unknown. It is assumed that it has been discovered and rediscovered all throughout history. This would explain the different names such as the golden mean and phi. It is believed that the golden ratio has been used by the Egyptian when constructing the pyramids and by the Greeks when constructing the Parthenon.  Euclid linked this ratio to the pentagram as well.

What makes the golden ratio so interesting is many artists don’t know about it but continually will make it in their work. People seem to find pleasure in that specific ratio. Many people have created art, of all kinds, and not realized that their masterpiece consisted of the golden ratio. One example is Fibonacci numbers. They have a strong relationship with the golden ratio and it is believed that Leonardo Fibonacci didn’t even know it. It is know that Fivonacci’s sequence goes for infinitely long. If two consecutive numbers of the Fibonacci sequence are divided by each other, the further in the sequence the closer it to phi (1.618). If looking at the first ten numbers of the Fibonacci sequence we have 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55. If we divide the number next by the previous number we will start to move closer to phi. This is shown in the table below

First number (F­a)

Number Next (Fb)

Fb/Fa

1

1

1/1=1

1

2

2/1=2

2

3

3/2=1.5

3

5

5/3=1.667

5

8

8/5=1.6

8

13

13/8=1.625

13

21

21/13=1.615

21

34

34/21=1.619

34

55

55/34=1.6176

 

Basically f(x) = Fb/Fa approaches phi (1.6180339) as x goes to infinity. It is really funny that a mathematician would find a sequence of numbers that would be ground breaking stuff and within this amazing work lies another more subtle amazing creation. So that leave me and other with the question why is this golden ratio so loved by the human eye and occurs time and time again in history?   

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