The Problem with 33

   

     In this weeks installment, the topic of the post will be on a problem concerning the number thirty three. This is a simpler problem, with not a large amount of backstory so the focus will primarily be on the problem itself, and not its contributors. This is partially due to the fact that there is limited information on the founders. In order to compensate for this the founders of the math used in the problem will be covered. That will be covered secondly  The first part of the post will deal with the problem itself. 

     The problem of 33 seems simple on the outside, but it has rather complex working behind it. This is because the problem stems from number theory, which deserves many blog posts about by itself, but will be mentioned briefly in this one. The problem starts out with a list, or sequence of numbers. The sequence stars out with one, then two, then three, then skips 4 and 5. There will be several numbers that are skipped that seem a bit random, but this will be addressed later on. The sequence continues without being interrupted, until it skips thirteen and fourteen, then it skips twenty two and twenty three, and finally it skips thirty one and thirty two. The sequence does continue on, but the sequence has reached thirty three, the important part of the problem. All the numbers on this sequence are thought to be sums of three cubic whole numbers. Meaning that they can be expressed as the sum of three whole numbers raised to the third power. In math terms the numbers on the sequence can be written as equaling a^3 + b^3 +c^3.  This is a Diophantus equation, and has been brought up in a previous post about Fermat's last theorem. In this particular case the negative values of the numbers are also allowed in order to allow for subtraction. In order to illustrate this the example of the solution to the number three will be provided. Three can be written as one cubed, plus one cubed, plus one cubed, or 1 + 1 + 1. This may seem simple, and many of the smaller integers do have simple solutions that can be worked out, but some numbers on the sequence have very large number solutions. It was not until 1999 when the solution for the number thirty was confirmed by the use of computers. The solution to the number thirty is, (2,220,422,9320)^3 + (-2,218,888,517)^3 + (-283,059,965)^3. This seems like a rather large sum of numbers, so it shows the scale of some of the answers to this sequence. 

     This is where the number thirty three comes along. Thirty three does not yet have a solution. Not only is the solution not know, whether there is a solution is also unknown. Not being able to determine if the number in the sequence can have a whole number solution to the Diphantus equation is irregular. So far it is know to all numbers up to 10^14 can be proven to have a solution or not have a solution. Now backtracking to the sequence itself there where a few numbers that were skipped. This is because they were proven to not have a solution to the Diophantus equation. It has been proven that if the number on this sequence can be written as nine multiple by a whole number plus four, or plus five, then it can not be written as a sum of three cubes. In mathematical terms if the number can written as 9n + 4, or 9n + 5, with n being a whole number, then it can not be a sum of three cubes. This has been proved, but the proof will not be covered in this post. So the number four was skipped because it can be written as 9(0) + 4. That is the problem concerning the number thirty three.

     As mentioned previously, this has to do with a branch of math called number theory. Number theory is the study of integers. Many of its contributor's work can be seen in this problem, or in work closely related to this problem. Diophantus equations are the biggest ones, but the work of Euler and  Gauss, with their summations can be seen.  Also the work of Ramanujan can be seen in the partitions of this problem, partitions being the ways you can break down number into sums and multiples. This means that even though this problem seems rather insignificant it has many ties to great mathematicians, and great math.

      Sources https://www.youtube.com/watch?v=wymmCdLdPvM
                    http://www.mathpages.com/home/kmath071.htm
                    

The Good Will Hunting Problem

This installment about mathematical problems is going to be on the problem from the movie, “The Good Will Hunting”. The focus will first be on the story behind the actual good will hunting, and secondly on the problem illustrated in the movie. The problem in the movie does not require an extensive amount of work to solve, but it seems to be a more popular problem that anyone could take a crack at. The hope is that this post will be more entertaining, and a little bit easier to read then some of the other posts made previously.

Will Hunting is more of an urban legend then an actual person. There are however several candidates, but there are two main candidates for the actual Will Hunting. One of them is thought to be George Dantzig. One day, while in university, George was late to class. He got there sat down and copied some problems down from the board, these were homework problems related to the lecture. George took them home, and he thought that the last two were rather hard problems, but he worked them out anyways. He turns in the assignment, and nothing happens for six weeks. Then early in the morning he gets a knock on his door, and it is his professor. The professor explains to him that he was only supposed to do the first few problems, the last two were unsolved problems dealing with statistics. George had solved these problems as a part of his homework. Later, when George was completing his PhD, he asked his professor what to write about, and he told him to put those two pieces of paper in a binder, and call it good. The second Candidate for Will hunting is a man by the name of William Sidis. William was a child genius. He was said to have been giving lectures at Harvard at the age of 11. He had a natural ability with math, but he had trouble with the law. He attended communist rallies and assaulted an officer. This lead him to be put in jail. He was released from jail under the condition that he saw a therapist. His therapist being with father. When he was finished seeing his father, he described it as a being a living hell, and swore to be done with academia. William Sidis never worked in a major academic position again. He found office jobs, and in particular he liked running adding machine, because they fascinated him. Though these are the candidates for the real will hunting the movie is not actually based on either, in fact the movie is based on a mathematician by the name of Ramanujan. Ramanujan deserves his own blog post, so in this post he will be covered briefly, but much like Will in the movie he was a math prodigy. Ramanujan was born in India, and his work was impressive considering where he came from. Ramanujan did not have a formal education. When he was in his twenties his potential was seen, and he attended Cambridge. Sadly he died soon after. Much of his work was done concerning partitions. Partitions are the ways you can break down numbers into multiples, and additions.

In the movie, “Good Will Hunting” an MIT professor presents a problem on the board. He claims that it took MIT professors two years to solve the problem. This is probably an exaggeration made for the film, because it is not that difficult to solve. The hardest part about the problem is to understand what the problem is asking. The question is to draw all the homeomorphically, irreducible, trees of size n=10. Breaking that down one piece at a time, trees refers to a network of lines and dots without cycles, meaning no closed polygons. Homeomarphically means you can stretch, and rotate a network tree but it still counts as the same trees. In doing so it puts a finite number on the number of solutions. Next, irreducible means that if there is a tree with a point that could be removed, and not change the tree, then the point should not be there. This is because it could be reduced, so the point was serving no purpose. The last part, n=10, means that the trees must have ten points, that need to be connected with lines using these parameters. There will be an example of one of the solutions in order to fully illustrate the problem at hand. There are ten trees that match the criteria explained, and there will be a link to the ten solutions.

All solutions http://25.media.tumblr.com/tumblr_lm8kawOpTO1qcj2hco1_400.png

Sources https://www.youtube.com/watch?v=iW_LkYiuTKE
              https://www.youtube.com/watch?v=SzjdcPbjaR4
              http://www.eoht.info/page/Good+Will+Hunting+(William+Sidis)