The Josephus Problem

This week’s mathematical installment of intellectual nourishment will be on the Josephus Problem. The Josephus Problem is not a rigorous problem, nor is it hard to solve, but it is a nice easy problem. This is to act as a break from the advanced problems that have been posted recently. Due to the format of the problem, and how it is set up, first the backstory of the problem will be covered, followed with the proving the problem, and finally a nice trick with binary.

The problem arises from a historical story. The scenario is there are Jewish soldiers that have been surrounded by the oncoming Roman army. The Jewish soldiers did not fancy capture, and would prefer death. A plan was devised that the Jewish solider would gather in a circle, and one solider, in what would referred to as position one in the circle would take his sword and kill the solider next to him, and the next living solider would kill the next living solider new to him. This process would continue until only one was remaining, and he was supposed to commit suicide. Josephus however was not a fan of dying, and would much prefer living. So the problem for Josephus is where does he stand in the circle to be the last one standing. In his case there were 41 soldiers in the circle.

To best illustrate the proof, lets start with a smaller circle. To best imagine this, the soldiers are standing in a circle the top of the circle is position one, and then it goes around just like a clock would.  In order to save time I have compiled the winning seat from a couple of circles. So if there is one individual, position one wins, with two people position one wins, with three people, number three wins, with four is one, five is three, six is five, seven is seven. From this small bit of information something can be derived already. The winning seat is always an odd number. This can be seen, because on the first pass through of killing all the even numbers go first. At this point is might seem like the pattern is 1,13,1357,13579,13579... but it is not. According to that pattern if there were thirteen soldiers, position one would win, but that is not the case, in fact position 11 would win. If you do not believe me try it for yourself. So if the pattern is not to go up by two each time, and then cycle back to one, what is? If you notice all the powers of two go to one. That can be easily proven if we know that all the evens go out in the first round. So if there is a power of two number of soldiers, it will get cut in half the first round, and position one will start the second round, and then will half, and half until it cannot be halved again. So the pattern would go 1,13,1357,13579(11)(13), 1...but how does that help us? We know that if there is a power of two number of solider at the beginning of a round then the first position wins. So if the biggest power of two is taken away from a number, the remainer would tell how many position to offset from one to get the winning position. For example, if there where seventeen soldiers that would be the same as sixteen plus one soldiers. So after one kill there is sixteen solider remaining and whoever's turn it is to kill would win the round. So whose turn is it? In the case of seventeen it would position three's turn. Going back to our data set, if there where five soldiers, or four plus one soldiers then three would also win. So if one is the remainder of one from our power of two subtraction, then three wins. So three times the remainder of the power of two, plus one would be the winning seat. Just to be sure lets try it if there were seven soldiers, since we already know the answer, it will allow us to confirm our results. The highest power of two that can be subtracted from seven is four, which would leave us with three. Three times two, plus one equals seven. This matches our results, and confirms it. Going back to the original problem, there were 41 soldiers, 32 is the greatest power of two that can be subtracted from 41 with remainder. 41 minus 31 is nine, nines times two is 18 add now is 19. Josephus should get in position nineteen of the circle.

Something that I will not prove, but might be interesting, if you take the number of soldiers and write in binary, to find the winning seat all you have to do is shift the leading number to the end, and reevaluate the binary.

Sources https://www.youtube.com/watch?v=uCsD3ZGzMgE
              http://www.geeksforgeeks.org/josephus-problem-set-1-a-on-solution/

The Four Color Problem

This week’s installment of mathematical problems is going to be about the four color problem. This problem has to with the involvement of filling in maps with different colors. The problem started with physical maps of countries and boundaries, and then shifted into a more abstract mathematical field. The problem will be covered first, followed up with a five color solution proof, and finally some history.

The four color problem is, if there is a map can all the countries or states can be filled in with four colors, in that no two same colors come into contact with each other. This was purposed in the mid 1800's and since then people have been filling in actual geographical maps with four colors. No matter the complexity of the geographical map, it will not require more than four colors to fill in all the boundaries. The map of the world can be done, the map of the Untied States can be done, and so can the map of the United Kingdom counties. All geographical maps fit these criteria. This is where the realm of mathematics steps in. Instead of using maps that already exist, can a map be created that can disprove this conjecture? This is because for the average person, it is a lot simpler to come up with a counter argument rather than proving it. So people started creating maps, and filling them in. There arose a problem of repeating maps being checked. Though two maps look different if the share the same type of relation between borders, then they are in essence the same map. So instead of filling areas on maps countries were represented with colored dots. If two countries touched reached other, they were connected with a line. This allowed for redundancy to be removed, and allowed for deeper analysis. This deeper analysis comes from the fact that the problem has been shifted from a coloring book problem, to a networking problem. In the 1970's this problem took off with the newfound interest in computer science, that arose in the time period. There is some rather useful insight about networking that comes in handy in solving the problem. If there is a country, call it country x, country x can be connected up to five other countries. Meaning in a given map every country can only connect with five other countries and still fit the parameters of a map. That is a very useful bit of information to have, because it allows the problem to simplified, and narrowed down. This will be apparent in the proof of at least five colors can be used.

In the last part it was mentioned that a country must have a network with anywhere from one to five connections in it. That part has its own proof, which was not covered. Going off of that information to prove five colors, start with seven colors, and work down to five. If there is a map with seven colors, assuming it is the smallest map with seven colors, there has to be one of those country x's that follows that networking rule. If that country is removed, or pulled from the map, then there is a smaller map. The smaller map would only require six colors then, and if the country x is put back into place, now there is an extra color that country x can now take, reducing the number of colors needed. This process can be done down to five, though five requires a tad more information and proofing, but can be done using the same logic.

In the mid 1800's a map creator was coloring in the counties of Great Britain. He tried to optimize his map by filling it in with as few colors as possible, without the same color touching. He worked at it, and did it with four colors. He assumed that four colors might be enough to fill any map then. He purposed the idea to his brother, who in turn passed the idea onto a mathematician by the name of Augustus De Morgan. Augustus is a famous mathematician, who is responsible for putting the problem together, and getting it some attention. The problem remained unsolved for one hundred and twenty five years after its proposal. Kenneth Appel and Wolfgang Aachen did not solve it until the 1970’s. There proof was controversial because it was the first mathematical proof to be done by the use of computer power. Which at the time was a not considered a guaranteed source.

Sources https://www.youtube.com/watch?v=NgbK43jB4rQ

                  http://www-groups.dcs.st-and.ac.uk/~history/Biographies/De_Morgan.html

                  https://www.youtube.com/watch?v=laMkuPrad3s&feature=youtu.be

The Importance of Pure Mathematics

            In today’s world of mathematics, there are two main fields of mathematics. Those fields are pure, and applied mathematics.  Pure mathematics in its simplest definition is math purely for the sake of math. Pure mathematics would include developments is number theory, Algebra, Control theory, and Calculus analysis. Applied mathematics in its simplest definition is math that has a purpose outside math itself. Applied mathematics would include math done in, physics, chemistry, biology, and any other math intensive science. For the most part pure mathematics is not seen as an important part of society. Sometimes pure mathematics is given the label of useless, or pointless. Although it might seem like a waste of time to some people, pure mathematics has an immensely important part, in the everyday workings of society as a whole.

            Pure mathematics is important because it develops the tools for applied mathematics. The groundwork of all Calculus, Algebra, Topology, and mathematics as a whole, rely on pure mathematics. Farida Kachapova wrote in the Journal of Mathematics and Statistics that, “ . . . without fundamental mathematics there would be nothing to apply.”(Kachapova) This means that all of the applied mathematics need pure mathematics to facilitate them. For example, a huge chunk of Newtonian physics relies on the understanding of basic algebra.   G. H. Hardy was once quoted saying, “Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.” (Hardy). What Hardy is saying is that the tools used for mathematics are more important then the products of mathematics. The tools of mathematics can be used to make all the results, while the results only help with that specific issue. That is why pure mathematics is important to applied mathematics.

            Pure mathematics is, and was used to develop many of the technologies used in the daily life of individuals.  The most commonly mentioned invention, which came about because of pure mathematics, is the computer. This is because the computer is revolutionary to society as a whole. Phones, laptops, Gameboys, and any other electronic available all rely on pure mathematics. Ben Orlin wrote about the invention of the computer saying that,” . . . one of the purest mathematical enterprises ever undertaken . . . It gave us the computer, which in turn gave us… well… the world we know (Orlin).  The invention of the computer changed the world, and it came from pure mathematics. Not only does the processing of the computer use pure mathematics, but also the components powering it use pure mathematics. Electrical engineering requires the use of imaginary numbers. For the longest time imaginary numbers were thought to be useless, it was thought that imaginary numbers meant nothing. Jim Lesurf wrote regarding complex numbers in electronics, saying that,” Complex numbers are used a great deal in electronics. The main reason for this is they make the whole topic of analyzing and understanding alternating signals much easier. “(Lesurf). This shows that even a pure mathematical idea can be used in today’s world. In understanding complex numbers, it allows for a better understanding of electronics, which are an integral part of technology today.

            Pure mathematics is an important part of society today, because pure mathematics is still relevant, and will always be relevant. Farida Kachapova also wrote that,” . . . even the oldest known mathematical formulae . . . known 2400 years ago by Babylonians, Chinese and later the Greeks …are the bread and butter of present-day elementary mathematics.” (Kachapova). This means that pure mathematics is still used today. The math the Greeks were using is still being used in the classroom today. Now not only is pure mathematics relevant today, but it will be relevant tomorrow, and for the future to come. The universe will never be fully explained, and there will always be new mysteries, new problems, and a need for new mathematics to try to explain it. Going back to G. H Hardy, “For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.” (Hardy). Pure mathematics will always be the tool of applied mathematics. This meaning that no matter whatever problems arise, pure mathematics will be used. The fundamentals of math will always be used to understand new forms of applied mathematics.

            Pure mathematics is often times seen as non-useful form of mathematics. This is simply because pure mathematics is math for the sake of math. The common misconception is that there are not any benefits from working in pure mathematics, and only applied mathematics gets real world results. Pure mathematics is the root of all applied mathematics. Any applied mathematics goes back to number theory, algebra, or calculus roots. Although pure mathematics might not be making the break through, or discoveries, it is facilitating them. Pure mathematics is incredibly important in society today. This is because pure mathematics is the fundamentals of all mathematics, pure mathematics helped developed many of the technological advances of today, and pure mathematics with always be relevant in mans quest of understanding the universe.

           

      Works Cited

Kachapova, Farida. "On the Importance of Pure Mathematics." Science Publications. N.p., 2014. Web. 15 Mar. 2017.

Lesurf, Jim. "Complex Numbers." Complex Numbers. University of St. Andrew, n.d. Web. 18 Mar. 2017.

Orlin, Ben. "Why Do We Pay Pure Mathematicians?" Math with Bad Drawings. Word Press, 25 Feb. 2015. Web. 18 Mar. 2017.

Unknown. "Quotations of G H Hardy." History.mcs.st-andrews. N.p., Dec. 2013. Web. 12 Mar. 2017.

Squaring The Circle

            This week’s post is going to be about a problem called squaring the circle. The problem dates back to the time of the Greeks. The problem itself is not the difficult part, the difficult part of the problem is that one of the restrictions of the problem is to solve it only using the technology available to the Greeks. First, this post will cover the problem itself. Secondly, the post will cover how pi relates to the problem. Finally, the post will cover how pi was proven to be transcendental.

            Squaring the circle is a problem about finding a circle with the same area as a square. At first this problem may not seem hard, but not impossible. After a few minutes of thinking, it might become apparent that a circle of radius one, and a square of side length root pi would have the same are. That would be correct, but the Greeks did not have the square root of pi. The difficult part of the problem is that, the problem can only be solved with the abilities of the Greeks. That means there is no algebra, there are straight edges, and compasses. Straight edges, and compasses limit the ability of available math. With this Greek Technology the four basic functions of mathematics, addition, subtraction, multiplication, and division, can be performed. Addition and subtraction are straight forward. Adding is putting two line segments together, and subtraction is removing a line segment. Multiplication, and division require some scaling with similar triangles, but are also possible. There is one more operation that can be performed with a straight edge and a compass, and that is square rooting. Imagine a line of length x, and then a distance of one is added to it, then a semi-circle is drawn to the end of the line. If a line is drawn vertically from where the line x and line of length one meet, up to the semi-circle, the length of the line is the square root of x. Looking back at one of the solution to the problem, one might ask then why could the Greeks not square root pi? The answer is that pi is not a constructible number, a constructible number refers to a number that can made with a compass, and a straight edge.

            Pi is a rather important number in mathematics, but at the same time it is equally complex, and mysterious. Pi is a transcendental number. The name for transcendental comes from the idea that pi is beyond other numbers, that it has transcended to a high plane of understanding. Or in other words, it cannot be expressed as an algebraic number. This is why the Greeks could not solve the problem. Since pi cannot be an algebraic number, then automatically it cannot be a construable number. However, pi was not known to be transcendental until the late 1800’s. Even though pi was known, and could be calculated, it was still unclear whether or not it was an algebraic number. It was proven to be a transcendental number by a combined understanding from Euler, and a proof of e’s transcendentalism.

            Two important facts about the number e were proven, e is transcendental, and e to any algebraic term is transcendental. This was proven via contradictions. What that means is that if e was algebraic, certain properties would apply. Since those properties do not apply, e is transcendental. Now the same thing is done with pi, but there needs to be a relationship between e and pi. This is where Euler’s Identity comes to the rescue. Euler’s Identity is e raised to the (pi) x (i) is equal to negative one. Going back to the proof of e raised to an algebraic power, if pi is algebraic then e raised to it, should result in a transcendental number. Negative one is not transcendental, so by contradiction it must not be algebraic, so if pi is not algebraic, then it is transcendental.

            Going back to the problem itself, this means that there was not a way for the Greeks to find pi. Even if the Greeks tried to use a square with a side length not having pi in it, the radius circle would have to multiple by a ratio of pi to get a whole number answer. Pi is crucial to this problem, and so it was proven that under the given rules of the problem, Squaring the circle is an impossible problem to solve.

Sources https://www.youtube.com/watch?v=CMP9a2J4Bqw
              https://www.youtube.com/watch?v=seUU2bZtfgM
              http://mathworld.wolfram.com/CircleSquaring.html