Backgammon

Let's take a look at one of the oldest board games to be played by two players: Backgammon. It's a game that involves strategy and a bit of luck from the roll of the dice to determine who wins. Backgammon is played by two people, where the board itself consists of twenty four triangles called "points" divided up into four quadrants of six triangles in each. The quadrants are the player's home board and outer board, and the opponent's home board and outer board. The home and outer boards are divided by a bar in the center, splitting it into a left and right side as you can see here:

The objective of the game is to move all your pieces, whichever color you are,  to your home board and then take them off the board in the process called "bearing off". The image above is the standard setup for every Backgammon game. You each have fifteen pieces placed in specific locations to start in every game and it's always those same spots. To win, you have to move all of them into your home board as indicated above. So if you are red, you have to move your pieces to the area labeled your color's home board and your opponent is doing the same as well. The direction you move the pieces depends on where your two pieces are on the board. In the above image, red has the two pieces on the bottom right, so you would move the red pieces clockwise, going through each quadrant to reach the color red's home board. For white, the pieces would be moving the opposite direction as indicated here:

To start,  each person rolls one die and whoever rolls higher goes first with the move from the two dice rolled. For example, if I roll a 6 while my opponent rolls a 2, then I go first and make my move of  6 and 2. Moving around the board is simple: you move one piece per each die you rolled or combine the roll to move a  piece. If you choose to combine the roll, let's say you rolled a 2 and a 5 so you would move a piece 7 spaces ahead, you would move the piece by each die you rolled. In other words, you can't jump ahead 7 spaces without moving 2 spaces first and then 5 or vice versa as well as the condition that your opponent doesn't occupy either space ahead. If you rolled doubles, then it's twice the rolls. If I roll two 5s, I can move four of my pieces five spaces ahead or combine. You can move any piece ahead unless the space is blocked by your opponent. A space on the board is blocked when your opponent has at least two pieces on the space. However, if your opponent has only one piece on the space, then you can attack their piece. Attacking results in your opponent's piece being knocked onto the bar of the board where it has to start from the beginning while your piece takes its spot.  This is the part where strategy comes into play. Leaving a piece alone by itself makes it vulnerable from being attacked by your opponent.

If you get your piece attacked and knocked onto the bar, then when it's your turn you must roll that piece back onto the board before moving any other pieces. It would sound simple getting the piece back onto the board but there's a problem with that: the quadrant where you bring the piece back in is also your opponent's home board. This can play as a strategy by your opponent when he gathers his pieces in the area later on in the game. There could be some spaces occupied by your opponent which would result in you having to roll a specific number to get the piece back in. Even worse, it's possible to be unable to move your piece back onto the board if all the six spaces are occupied by your opponent's pieces which result in you wasting turns until one of the points becomes available.

Bearing off can only happen once all of your pieces are in your home board. When it's your turn, you roll the dice to bear off each piece. In your home board, the furthest triangle is 6 spaces away to bear off, then the next is five spaces away, and so on. You can only bear off pieces that are in that space that matches the dice roll. For instance, if I roll a 4 and a 3, then I move a piece from the point that's four spaces away to bear off and a piece from a point that's three spaces away. You win when you get all of your pieces off the board first.

The math behind this game is more on the lines of probability since you are using the dice to move around the board. In an opening move there are certain moves that you can a make based on the roll you get whether you went first or second. It also comes down to the probability of getting a certain number you want to try to get ahead in the game. In a scenario where your opponent occupies four points and you're trying to get a piece onto the board, it would be simple to calculate it. You would multiply the chance of not entering for each die as they are independent of one another, then subtract that result from 1 as doing that would give you the probability of entering. You would do this for each scenario and end up getting all of the probabilities for each one. The example where your opponent has four points occupied can be illustrated here to figure out the chances of entering your piece back in:

p(not entering) = 4/6 x 4/6 = 16/36,   p(entering) =1 - 16/36 = 20/36

There are different strategies you can use to win Backgammon. The Running strategy involves moving your pieces as fast as possible based on the roll of the dice and is ideal if you start out with high numbers in the beginning. The Blitz involves attacking and sending their pieces onto the bar which would slow them down as they would have to try to roll them back in. Holding involves blocking points to make it difficult for your opponent to bring them back in due to a lesser chance they would have based on those probabilities discussed earlier. This is the basic game to play without getting too technical as it would take more than 1,200 words to completely explain the concept of the doubling cube and preferred opening moves. From experience playing this game, I'd highly recommend it as something new to try out.

Bibliography

Brown, Seth. "5 Simple Strategies to Win Backgammon." About. About, 2014. Web. 20 Nov. 2014. <http://boardgames.about.com/od/backgammon/a/Basic-Backgammon-Strategy.htm>.

Keith, Tom. "Backgammon Rules." Backgammon Rules. Backgammon Galore, 2012. Web. 20 Nov. 2014. <http://www.bkgm.com/rules.html>.

Packel, Edward. The Mathematics of Games and Gambling. Washington: Math. Assoc. of America, 1977. Print.

Art Gallery Theorem

The “Art gallery theorem” or “museum problem” as it is referred to, was originated from the real world situation. There are x amount of guards guarding the gallery, and the goal is to compute the minimal amount of guards needed to guard the entire art gallery. The art galleries are typically given the shape of a polygon. And “a polygon is generally defined as an ordered sequence of at least three points v1,v2…,vn in the plane, called vertices, and the n line segments v1v2, v2v3, …, vn-1vn, and vnv1, called edges.” As for a “simple polygon” it is essentially the same as a polygon but except for the fact that non consecutive edges do not intersect. A simple polygon is also known as a jordan curve which means that it is divided into three subsets, the interior, the polygon itself, and the exterior. What most mathematicians take as their first step is the decomposition of the polygon. And there are many ways to decompose a polygon as the first step, for example there is, “convex polygons”, “spiral polygons”, and “monotone polygons”. Yet when dealing with a star shaped polygon it is shown that one guard is needed to cover the entire polygon when place in the fixed position.

   

The art gallery theorem was brought up within a conversation between “Klee” and “Chvatal”. This theorem was proposed to calculate the minimal amount of guards needed to cover any polygon. As Chvatal began to work on the problem he began to realize that g(n)=[n/3] (n= number of walls), and this was the way to find the minimal amount of guards needed. Later on “Fisk” gave a concise proof stating that you needed to triangulate the polygon then assign each vertex one of the three different colors. So this way it prevented any adjacent vertices from having the same color. What has also been stated about the art gallery problem is that based on the placement of the guards which was stated by both Avis and Toussaint. They both believed that there was a set of guards that were needed to cover the polygon yet that there was an algorithm to find the placement of each guard.

When approaching the art gallery theorem most first steps is to make the polygon into a polygon triangulation (if not possible try other decompositions based on the shape). This means that the polygon is broken up into triangles which are generated by the diagonals within the polygon. There are also different ways of guarding the polygon. For example, there is a “covering guard set” and a “hidden set”. The difference between the two is that in a “covering guard set” each guard is able to see one another. As for in the “hidden set” each guard is placed in a spot of the polygon where they are incapable of seeing one another.

    

When dealing with orthogonal comb polygons Kahn, Klawe, and Kleitman showed that the maximum of guards needed was the way solving such a polygon “orth(n)>[n/4]”. They also took the idea of triangulating the polygon but tweaked it a bit by dividing the polygon into “convex quadrilaterals”. They also used four colors instead of three, and this caused there to be all four colors at each vertex. This is an example of one the polygons in which needed to use a different decomposition instead of just triangulating the polygon.

Given information that has been proven is that when n=5 or less one guard will be needed to cover the polygon. Another given hint is that in some 6-vertex polygons there are only two guards needed when placed in a fixed position.

Works cited:

• Shermer, Thomas C. "Recent Results in Art Galleries." Recent Results in Art Galleries (geometry) - Proceedings of the IEEE (n.d.): n. pag. Sept. 1992. Web. 17 Nov. 2014.
• Avis, David, and Godfried Toussaint. "An Efficient Algorithm for Decomposing a Polygon into Star-shaped Polygons." Pattern Recognition 13.6 (1981): 395-98. 9 Dec. 1999. Web. 17 Nov. 2014.
• O'rourke, Joseph. "ART GALLERY THEOREMS AND ALGORITHMS." (1987): 1-296. Oxford University Press. Web. 17 Nov. 2014.

1 Is Not a Prime Number

It is a very common misconception that the number 1 is a prime number. If a prime number is a number with factors only of itself and 1 then why is 1 not a prime number? 1x1 is the only factor of the number 1 so therefore the number 1 must be a prime number. Many mathematicians in the past even believed 1 to be prime. This is wrong. The real definition of a prime number is “An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself.” (Caldwell). The key words in that phrase are “greater than one”. But this does not answer the question why 1 is not a prime number.

There exists a really important theorem, by Euclid, in mathematics that can help us prove that 1 is not prime, the fundamental theorem of arithmetic. It states, “every positive whole number can be written as a unique product of primes”. (Grime) This theorem helps us understand that primes are the building blocks for all numbers. Lets take the number 15 as an example. 15= 3x5. 3 and 5 are prime numbers. Now look back at the theorem, there is a key word in understanding the theorem, and that word is unique. Since it has to be a unique product of primes, that means there is only one way in which we can write the product. Now if 1 was a prime number we could state that 15= 3x5x1, we could also write it as 3x5x1x1, or we could write it as 3x5x1x1x1. If one were a prime number there would not be unique ways in which to write positive whole numbers as products of primes.

The number 1 is a unit.  Euclid states in his VII book of elements definition number 1. “A unit is that by virtue of which each of the things that exists is called one” (Heath, 277). This definition of a unit is quite hard to comprehend. Heath helps us with the understanding of each definition through further explanation. “The etymological signification of the word μovás (Greek for the word unit) is supposed by Theon of Smyrna (p. 19, 7-13) to be either (1) that it remains unaltered if it be multiplied by itself any number of times, or (2) that it is separated and isolated from the rest of the multitude of numbers.” (Heath, 279). Even though many people once believed 1 to be a prime, it is the significance of units in modern mathematics that causes mathematicians to be much more careful with the number 1.

          Grime, James. "Why One Is Not Prime - Video - Numberphile - Videos about Numbers and Stuff." Why One Is Not Prime - Video - Numberphile - Videos about Numbers and Stuff. N.p., n.d. Web.

Caldwell, Chris K. "Why Is the Number One Not a Prime?" Why Is the Number One Not a Prime? N.p., n.d. Web.

Heath, Thomas L., Sir. The Thirteen Books of The Elements. Vol. 2. Toronto: General, n.d. Print. (Books III-IX).

Applications of Mathematics in Basketball

Mathematics plays a huge role in basketball, from the shooting percentage to how much time a team has to score in. An article written by Brian Skinner talked about the probability that a given shot will do in, equality of future shots the team is likely to generate, and the number of seconds left before the players must either shoot or forfeit the ball to the opposing team. Skinner’s equations show that if two teams shoot with different times left on the clock that the team that shoots with extra seconds has a better shot selection which turns into a winning strategy.

           

Professional basketball player Lebron James is now apart of the new Big 3; himself, Kevin Love, and Kyrie Irving. James can and will score from anywhere on the floor. His highest scoring spots are in the paint and is average outside of it. Although, he still leads the league in close-range efficiency with 75 percent of them are inside eight feet.

This shows how much he scores in the paint compared to anywhere else on the floor. He doesn’t just score from inside the paint, although his scoring percentage is at it’s highest in the paint, he does score from mid-range shots and some three-point shots. With percentages of 47% mid-range shots and 41% three pointers. James is a stronger scorer on the left side of the floor compared to the right.

            Another professional basketball player whose name is Kevin Love also enjoys scoring from the left side of the floor. He is by far NBA’s most active shooter form the three-point line on the left side.

This diagram shows that he is most active on the three-point line but can also score efficiently inside the paint as well. He is increasing in his ability to score from everywhere on the floor, especially behind the three-point line.

            The third component in the Big 3 is Kyrie Irving. He has become one of the go-to playmakers for Team USA. Irving has been one of basketball’s most creative scorers despite drama within the league. Unlike James and Love, Irving scores from all over the floor and is very efficient. He moves and opens up the floor for his team and himself, showing that like James and Love he is above average on the left floor but with his inside game along with his outside game.

With James and Love joining a team with Irving, his usage will begin to diminish which will lead him to take fewer self-created tough jump shots than his previous season. On the bright side he will have a higher efficiency with catch-and-shoot shots than off the dribble. So his chart will begin to look more orange and red which means he will be more above average on his shot selection.

            Having the Big 3 on the same team in Cleveland, they will become one of the NBA’s productive scoring teams. Although with the combination of the three, their inside game will be stronger which will leave the outside game for the other players on Cleveland.

The left side of the floor will be dominant for Cleveland. They will score a majority of their points from outside the three-point line and inside the paint. Scoring on the right side will have to come from the other players on the team according to this chart, the Big 3 loves the left side of the floor, so the team will have to balance it out unless only scoring from the left side will win them ball games.

           

            Within mathematics in basketball rebounding plays a huge factor. This article is based upon the question of “Where do rebounds go?” Last regular season of the NBA players missed over 100,000 shot attempts. Missed shots means rebounds, missed shots result in the outcome of the game. Rebounding is a huge potential to the game of basketball. Rebounding is very unpredictable, missed shots means that the ball can hit the rim or background and bounce anywhere.

            Almost 80 percent of all NBA rebounds happened within eight feet of the hoop. Thinking about rebounding should include three simple rules: distance matters, direction matters, there will be randomness.

            Referring to the distance matters rule, longer shots lead to longer rebounds is supported by basic physics. “The tracking system enables us to visualize the distance effect like never before.” The further out you shoot the longer the rebound. Within the paint, on average, the rebound will go about 5.4 feet from the basket. Within 18 feet from the three-point line and the lane, on average, the rebound will go from 6.1 feet to 6.5 feet from the basket. Finally within the three-point line, on average, the rebound will go from 7.3 feet to 8.3 feet from the basket.

            Referring to the direction matters rule, rebound locations depend on the angle the shot was taken at. For example, if a player misses a shot from the right corner, the rebound on average will come back in the same direction or will go on the opposite side of the basket. “Consequently, a basketball traveling from left to right will continue on its path unless it hits the rim, in which case the rim exerts a force on the ball.” The ball has to hit the rim in special ways to change its path of direction.

            Referring to the there will be randomness rule, some times a player doesn’t know which way the ball is going to land after hitting the rim on a shot. This rule includes the other two rules; distance matters and direction matters.

            In conclusion mathematics is very essential in the game of basketball and is very hard to explain in fewer than 1,200 words. There are many more characteristics to basketball that involve mathematics. The majority of the game is mathematics related but it was hard to pick out the most essential ones and put them into words within an essay.

References

Cowen, R. (2011, August 2). The Mathematics of Basketball. Retrieved August 2, 2011, from http://news.sciencemag.org/2011/08/mathematics-basketball

Goldsberry, K. (2014, January 1). The Shape of Cavs to Come: How Lebron, LOVE, and Kyrie Might Fit Together. Retrieved January 1, 2014, from http://grantland.com/the-triangle/lebron-james-kevin-love-kyrie-irving-cavs-offense/

Goldsberry, K. (2014, January 1). How Rebounds Work. Retrieved January 1, 2014, from http://grantland.com/features/how-rebounds-work/

Fractals

Fractals are described as “a geometrical figure in which an identical motif repeats itself on an ever diminishing scale (Lauwerier, xi).” Fractals are both man-made and found in nature, for example snowflakes. Fractals go beyond just patterns and have been utilized in technology and architecture. On that account, fractals can play several important parts in our daily lives.

Although fractal geometry only dates back to 1975, it is important to know that work on fractals started around the 17th century. It began to take shape when Gottfried Leibniz contemplated recurrent self-similarity. However, he made the mistake that a line was self-similar in this light. Famous mathematicians have added to this research like George Cantor. He created sample subsets with unusual characteristics called cantor sets, which are now recognized as fractals.  Pierre Fatou and Gaston Julia added work by mapping complex numbers and iterative functions, but it “led to no illustration in their time (Barnsley, 2).” Without the help of computers, these researchers were limited to manual drawings, impeding greater understanding for its beauty and its implications. This was changed by Benoît Mandelbrot, who helped piece together many years of hard work on the topic. He used computer visualizations to help solidify his definition of fractals, a word he also coined.

Many fractals can be found in nature. In the book “Fractals” by Hans Lauwerier, he provides an excellent example that is easy to perceive. Imagine a tree with a trunk that separates into two branches. In turn, these two branches separate each into two smaller branches, and so on. This can be infinitely repeated in our heads. “Each individual branch, however small, can in its turn be regarded as a small trunk that carries an entire tree (Lauwerier, xii).” This can be illustrated in the picture down below.

This construction has to do a lot with the binary system. The picture above, conceptually, has an infinite number of branches extending. To make a more simplistic-looking, and mathematically based tree, we would use only vertical and horizontal lines. First you start with a “T” shape, and each branch extending is a “T.” At every level the vertical lines would separate into two, scaling down by a ½ each time. These number of these vertical lines would double at each level. Moreover, the horizontal lines would be twice the length of the vertical segment below it. To start, if when began by giving the first line, the trunk, a length one, then the equation to continue on would be: 1x1 + 2x½ + 4x¼ + 8x⅛ +...

Fractals are also found in man-made structures. An example of this being used back in the day are old rope bridges. The Icans wove long lengths of stiff qoya grass by hand. These get woven into larger fibers, which then help create a stronger a sturdier rope. This same idea also applies to modern bridges like the Golden Gate Bridge. The giant steel cables are formed from a bundle of smaller cables, that, additionally, are constructed from smaller bundles. Another instance in where fractals are utilized in buildings and structures is the Eiffel Tower. On page 131 in Mandelbrot’s book “The Fractal Geometry of Nature,” he describes how “the tower that Gustave Eiffel built in Paris deliberately incorporates the idea of a fractal curve full of branch points.” He goes on further to explain that the tower is not made of solid beams but out of “colossal trusses.”  “A truss is a rigid assemblage of interconnected sub-members, which one cannot deform without deforming at least one submember. Trusses can be made enormously lighter than cylindrical beams of identical strength (Mandelbrot, 132).” You can see how the truss below is repeated through the structure. However, this wasn’t a new discovery that Eiffel had discovered. The idea that strength lies in branch points was exercised by designers of gothic cathedrals. As visualized in the Milan Cathedral, one can see the use of large arches interlocked

with smaller arches. The use of fractals helps give these buildings their beauty and strength.

An instance in where fractals are used in technology is in radio and antenna signals. A professor at Boston University, Nathan Cohen became inspired by the Koch Snowflake, a famous fractal construction, to create a more compact radio antenna only using wires and pliers. Today, antennae in cell phones use such fractals as a way to maximize receptive power in a minimum amount of space. The pattern’s compactness is what allows this strong signal.

In Geometry, fractals can generate amazing patterns and works of art. One type of fractal based in geometry is a “ternary” tree. We first start with a point and three protruding branches. Each branch creates a 120° with each other. Then each branch gets another three branches, ⅓ of the length of the previous one. This keeps on going to infinity until it creates this triangular looking snowflake.

In 1913, Polish mathematician Wacław Sierpiński created a nice variation of the ternary tree. It is obtained by starting with a solid equilateral triangle. The next step is to divide this into 4 smaller equilateral triangles by removing one in the middle. With the three remaining solid shapes, we repeat the same step and remove a triangle from each of them, making each triangle have 4 smaller ones. This is repeated indefinitely.

Looking at fractals at a surface level, it may seem that they serve no purpose beyond creating patterns and art. However, they can be exploited to created extraordinary things that help out in our world today.

Lauwerier, H. A. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton UP, 1991. Print.

Barnsley, M. F. Fractals Everywhere. Boston: Academic, 1988. Print.

Peitgen, Heinz-Otto, Dietmar Saupe, and M. F. Barnsley. The Science of Fractal Images. New York: Springer-Verlag, 1988. Print.

Mandelbrot, Benoit B. The Fractal Geometry of Nature. New York: Freeman, 1983. Print.