Gauss' Problem

            This week's post will be covering a problem solved by Gauss, and how it can be applied to another problem. To start things off I want you to add all the digits, of all the numbers from one to a million. Notice how it’s not the numbers, but the digits of those numbers that are being added up. So its not twenty-five plus twenty-six, it is two plus five plus two plus six. Think about this, and I will get to it at the end of the post. First the history of Gauss will be covered in their post. Then his similar problem will be solved. Finally I will connect the two and give the solution to the first problem I presented.

            Johannes Carl Friedrich Gauss was born in southern Germany in the late 1700's. Though he is not as famous as Newton, Gauss is still considered to be a great contributor to world of mathematics. His works included developments in number theory, such as advances in algebra, and statistics. He we also know for some of with work in the world of physics, as many mathematicians where in his time. He died in the mid eighteen hundreds.

         While in his early years Gauss was known to be a child prodigy. He seemed to have a natural ability with mathematics, and often times astonished people. One day while still in his youth Gauss went to school as he normally would. However, the professor was especially grumpy that day. Many variations of the story say he was old, and cranky, and would beat kids with his cane, making him seem more menacing then he probably was. The grumpy professor walked into the one room classroom, and since he did not feel like putting up with the children, he told them to add all the numbers from one to a hundred. At this time there was not pens and paper, but instead the children had blackboard slates. Now seconds after the professor had challenged the boys, Gauss runs up and puts his slate face down on the teacher’s desk. Thus submitting his answer. At this point in time Gauss is not the oldest boy in the room, and many of the other older boys look down on him. They were said to be thinking, poor little gauss, he has gone with a guess. Well time goes on, and all the other boys start working out the problem, using addition, multiplication, subtractions, and division, anything to try to get done first. Once the boys finished the problem they did like Gauss had previously done, putting their slate face down on the teachers desk, stacking them up. Once the time was up, the professor walked up to the desk and flipped all of the slates over. Now all the slates were in order from first to last, with the answers facing upward. Sitting on top of the pile was Gauss' answer. Gauss' answer was the correct answer, which is 5050. 

         So how did Gauss do it? Gauss, then explained, that the problem needs to be looked at in a different way. The most brute force way would be to add one plus two plus three, all the way to one hundred. Instead of doing it like that Gauss looked at in a different way. If all the number one through hundred are laid out one a number line, then Gauss' method can be seen. If you take one and add it to one hundred the answer is one hundred and one. The answer is the same if you add two and ninety nine. The same would apply to all the pair of numbers that can be created, all the way down to fifty and fifty-one. All Gauss had to do was take one hundred and one, and multiple by the number of pairs. In this case that would be fifty, so the answer would be fifty times one hundred and one. Fifty times one hundred and one is 5050.

          Going back to the problem at the start of post. Using the same process, start by writing out all the numbers from one to a million. Now notice that if you add to one to 999,998 it equals 999,999. This process can be done again and again. This process is the same pairing process gauss used. So then i would take the sum of all the digits of 999,999 and multiple that by the number of pairs. If zero is included that would make 500,000 pars. So 54 times a half a million would be twenty seven million. Not forgetting the one in one million would make the solution to the original problem 27,000,001.

Sources http://planetmath.org/gausscarlfriedrich

                  https://www.youtube.com/watch?v=Dd81F6-Ar_0

The Eight Queen Problem

          In this week's installment of mathematical insight, the topic will focus on the eight-queen problem. The post will start off with a small history, and backstory to the problem. Followed with the problem itself paired with some information on the scale of the numbers used in the problem. 

          The problem was originally published by Max Bezzel in the mid eighteen hundreds. Bezzel was a both a mathematician, and a chess enthusiast. It took two years for solutions to appear. Franz Nauk released them. Nauk then went to continue the problem into a more generalized state of understanding. The problem was modified from having eight queens to having "N" number of queens. After changing the number of queens Nauk modified the dimensions of the board. Instead of the traditional eight by eight board, it became "N" by "N" boards. "N" is not only the length, and width, but also the number of queens on the board. Mathematicians such as Gauss and Gunther have since contributed to the problem. 

         Looking primarily at the standard value of eight for "N", this meaning the board is a standard eight by eight chessboard, and there are eight queens on the board. The problem asks, if possible can all "N" number of queens be on the board and not be threatened by another queen? Color of the queen does not matter in this case, and if it can be done, how many possible ways can it be done? A queen in chess is considered to be the single most powerful piece, as it is the most versatile piece to move around the board. It can move both horizontally, vertically, and across diagonals. This makes positioning eight of them without threatening another difficult on a confined space. There are solutions to the standard eight by eight board. Starting with the total possible ways that eight queens can be positioned on a eight by eight board is the first part. There are sixty-four tiles on a standard chessboard. That means there are sixty-four factorial possible tile positions. If eight of the positions are taken up with queens that leaves fifty-six factorial tile positions. Since the queens are interchangeable among their given positions eight factorial also comes into play. So to find the number of ways that eight queens can be positioned on a chessboard, the total, sixty-four factorial, would be divided by 56 factorial multiplied by eight factorial. A factorial refers to that number multiplied by every previous whole number before that number. For example six factorial would be six, times five, times four, times three, times two, times one. A factorial has the nice property of showing how many ways you can arrange a number of times. For example if you wanted to know how many ways to arrange three items, the answer would be three factorial, or six. That makes numbers such as sixty-four factorial an immensely large number. Same with respect to fifty-six factorial. The net calculation on the number of ways to arrange the eight queens is roughly around four billion. This includes all permutations, but without the restriction of non-intersection between queens. Factoring that in reduces the number down to 92, continuing the sifting of possible outcomes, by removing redundancies due to reflections and rotations, the number goes down to twelve. There are twelve possible solutions to the eight-queen problem.

          To put factorials into scale of how big these numbers are, fifty-two factorial is less then the number of possible tiles, sixty-four factorial, and the number on non queen tiles fifty six factorial. Fifty-two factorial is the number of possible ways a deck of cards can be arranged. This is chosen because there is rather nice statistic about fifty-two factorial, and it is smaller then some of the numbers used in this problem. To understand fifty-two factorial, first stand on the equator. Start playing solitaire with a deck of cards, assume every new game is a new deck, and you win every game. Every billion years take a step forward, which should allow for a couple trillion games every step. Every time you walk around the earth remove a drop from the Pacific Ocean. Once the ocean is empty start stacking pieces of paper on the bottom. Once you have reached the sun, roughly one hundred million miles away. Notice not all decks have been played yet, so remove the papers one at a time, and put the water back in the ocean drop by drop, and go to and from the sun a thousand more times. Now about a third of 52 factorial decks have been played. That is how big fifty-two factorial is.

Sources http://www.datagenetics.com/blog/august42012/

                https://www.youtube.com/watch?v=jPcBU0Z2Hj8