Why you like Risk Management Plans Essay Example | Topics and Well Written Essays - 750 words

Why you like Risk Management Plans - Essay Example A hazard the executives plan is an archive that is made by the undertaking supervisor to gauge dangers, survey the effect of the dangers and characterize answers for the difficulties. Hazard the board designs additionally contain a hazard appraisal grid that decides the effect of a hazard over a particular period (Power, 2008). The method of making a hazard the executives plan begins with seeing how chance administration functions then the venture is characterized, expressing all the dangers and their conceivable effect. Contribution from different partners is significant at this stage. Every single superfluous issue are expelled then likelihood of each hazard happening is determined. The potential misfortunes from the dangers are then relegated with an effect number on a scale from 0.00 to 1 with less unsafe exercises having a little worth. Next, the all out hazard is figured then relief systems are created. From that point onward, the task director makes emergency courses of action and investigates the proficiency of the systems. At long last, a successful hazard is determined. A hazard the board plan has a few goals. The arrangement computes dangers and the likelihood of the dangers bringing about misfortune. This is significant while deciding if to seek after a business venture. A hazard the executives plan makes methodologies for overseeing dangers or dealing with the misfortunes from the hazard. The arrangement lessens the chance of amazements and supports effective utilization of authoritative assets (Rejda, 2011). Hazard the executives plans contain techniques on the most proficient method to handle dangers. These systems incorporate hazard shirking, misfortune decrease, and hazard control, spreading the hazard, self-maintenance, and duplication of assets, tolerating dangers and moving dangers. Hazard evasion is the most effective hazard the board procedure. By maintaining a strategic distance from a hazard, the chance of misfortune is wiped out totally. Hazard evasion is the best methodology however not the most viable one. It is difficult to take out all parts of hazard in an undertaking. A few

How Neologisms Keep English Alive

How Neologisms Keep English Alive A neologism is a recently authored word, articulation, or utilization. Its otherwise called a coinage. Not all neologisms are totally new. Some are new uses for old words, while others result from new mixes of existing words. They keep the English language alive and present day. Various components decide if a neologism will remain around in the language. Once in a while will a word enter normal utilization, said the author Rod L. Evans in his 2012 book Tyrannosaurus Lex, except if it reasonably unmistakably takes after other words.â What Qualities Help a New Word Survive? Susie Dent, in The Language Report: English on the Move, 2000-2007, talks about exactly what makes another word effective and one that has a decent possibility of remaining being used. During the 2000s (or the noughties,â oughties,â orâ zips), a recently stamped word has had a phenomenal chance to be heard past its unique maker. With 24-hour media inclusion, and the boundless space of the web, the chain of ears and mouths has never been longer, and the reiteration of another word today takes a small amount of the time it would have taken 100, or even 50, years prior. Assuming, at that point, just the littlest level of new words make it into current word references, what are the deciding components in their prosperity? Roughly, there are five essential supporters of the endurance of another word: convenience, ease of use, presentation, the toughness of the subject it depicts, and its potential affiliations or augmentations. On the off chance that another wordâ fulfillsâ these hearty rules it has an awesome potential for success of incorporation in the advanced vocabulary. When to Use Neologisms Heres some exhortation on when neologisms are helpful from The Economist Style Guide from 2010. Some portion of the quality and imperativeness of English is its preparation to welcomeâ new words andâ expressions and to acknowledge new implications for old words. However such implications and uses frequently leave as fast as they showed up. Before getting the most recent use, ask yourself a couple of inquiries. Is it prone to breeze through the assessment of time? If not, would you say you are utilizing it to show exactly how cool you are? Has it previously become a clichà ©? Does it carry out a responsibility no other word or articulation does similarly also? Does it loot the language of a helpful or popular significance? Is it being adjusted to make the scholars writing more keen, crisper, increasingly musical, more clear as such, better? Or then again to cause it to appear moreâ withâ it (indeed, that was cool once, similarly as cool will be cool now), increasingly grandiose, progressively bureaucratic or all the more politically right as it were, more terrible? Should the English Language Banish Neologisms? Brander Matthews remarked on the possibility that developmental changes in language ought to be denied in his book Essays on English in 1921. Notwithstanding the exacerbated fights of the upholders of power and convention, a living language makes new words as these might be required; it offers novel implications to old words; it gets words from remote tongues; it alters its uses to pick up explicitness and to accomplish speed. Regularly these oddities areâ abhorrent, yetâ they may win acknowledgment whether they affirm themselves to the lion's share. This unstoppable clash among security and change and among power and autonomy can be seen at all ages in the advancement everything being equal, in Greek and in Latin in the past just as in English and in French in the present. The conviction that a language should be fixt, that is, made stable, or as it were, prohibited to change itself in any capacity, was held by a large group of researchers in the seventeenth and eighteenth hundreds of years. They were increasingly acquainted with the dead dialects, in which the jargon is shut and in which utilization is frozen, than they were with the living dialects, where there is consistently perpetual separation and ceaseless augmentation. To fix a living language at last is an inert dream, and if could be achieved it would be a desperate disaster. Fortunately language is never in the selective control of researchers; it doesn't have a place with only them, as they are regularly disposed to accept; it has a place with all who have it as a primary language.

Paraphrase for Discussion Essay Example | Topics and Well Written Essays - 500 words - 5

Reword for Discussion - Essay Example For the situation where numerous cylinders are set in a stunned example, similarly as in the investigation, each extra column logically makes more disturbance on the other continuing lines. Along these lines, this expansion in choppiness prompts an expansion in heat move for each cylinder lines. A sharp gander at the pace of increment in heat move starting with one line then onto the next will uncover a diminishing pattern with more abatement acknowledged in those lines most distant from the primary column of the warmth exchangers. This is owing to the way that all the stream in the in the primary line of warmth exchangers is from the start laminar before being changed over into violent. Then again, the resulting columns work as a choppiness creating network, which in this manner builds the disturbance in the whole warmth exchanger. In any case, for each extra column experienced, a bigger level of the stream hitting a specific line is as of now violent before the hit prompting an arrival of only a little of level of the stream for transformation to fierce stream. For example, if 60% of laminar stream is changed over to violent for each column that the stream experiences, the profits will be reducing. This suggests for the primary line, 60% of the stream will be violent thoug h for the second and third lines just 78% and 84% individually will be changed over. At long last, the other factor that influences the warmth move coefficient is the distinction in temperature between the outside of the component and air. Since the warmth move coefficient changes contrarily with the distinction in temperature (see condition 5), this suggests move coefficient for a given warmth move increments as the temperature contrast diminishes. The subsequent investigation was directed to inspect the impact of warmth move as an element of area on a chamber. A round and hollow test was utilized to quantify the warmth move as a component of edge. The estimations were taken for uniform stream in an air burrow. A

About Young Goodman Brown - Free Essay Example

Sample details Pages: 3 Words: 965 Downloads: 9 Date added: 2019/08/02 Category Literature Essay Level High school Tags: Young Goodman Brown Essay Did you like this example? As dusk settled in the town of Salem, Goodman Brown kissed his wife, Faith, goodbye and commenced his journey toward the gloomy forest, an unholy errand he believed to be an exceptional one. While Brown paraded down the path, he expressed his concern about what may be lurking around and, as he continued, a mysterious man with a staff resembling a snake, approached him from nearby. Browns conscience told him to turn back, as this older man had a fatherly resemblance of both Brown and the Devil. Don’t waste time! Our writers will create an original "About Young Goodman Brown" essay for you Create order He then expressed that even Browns father and grandfather had walked this same path, as he helped them in their wicked ways. Not long after, Brown was amazed to see Goody Cloyse, the pious old women who once taught him his catechism, to seemingly be a witch. Brown was still confident that he could leave the forest, and as his companions departed, the lively staff was left behind as he, sat himself down on the stump of a tree and refused to go any further (Hawthorne). However, Brown discovered the most pious villagers, including the clergy members, passing on their way to a meeting where a young woman was taken into communion that night. Voices from Salem village, along with that of Faith, could be heard from the distance. Without thought, Brown rushed forth into the forest where he reached a gathering of witches sabbath, thus crossing over to the Devil. As a pair, Brown stood with Faith along what they believed was irredeemable wickedness. When evil prepared to baptize the newcomers into the deep mystery of sin Brown exclaimed to Faith, look up to heaven, and resist the wicked one (Hawthorne). Then suddenly, the scene had ended and Brown was abandoned in the forest alone, where we are left wondering if this witch-meeting was a dream or a figment of Browns reality. Nathaniel Hawthorne uses Young Goodman Brown to portray the Puritan Salems witchcraft frenzy that occurred during his ancestors time. Religion was very important to the Puritans of Salem village with the church being the political and social center of life. Law required all villagers to attend church every week and follow a strict moral code, with any violators said to be committing a sin against God. Because the Devil, chooses people who are considered to be the weakest in maintaining Puritan morals, Puritans were afraid that the Devil was always present, making the thought of witchcraft even more frightening (Religious Influences). These witches caused hysteria in Salem and continued to affect all citizens in the village, which ultimately broke trust between its people. During the trials of the sinners, Hawthornes great-great-grandfather, John Hathorne, served as a judge in the years 1662 and 1663. These people, many of whom were women and children, were deemed punishable under Pur itan Law and were convicted based on testimonials. During this time, 200 people were accused of practicing witchcraft and some were even executed (Blumberg). In the story, Hawthorne mocks the mass hysteria of the era by casting the entire village of Salem as devil worshippers. When Brown made his way toward the congregation, the Devil announced: There are all whom ye have reverenced from youth This night it shall be granted you know their secret deeds: how hoary-bearded elders of the church have whispered wanton words to the young maids of their households; how many a woman, eager for widows weeds, has given her husband a drink at bedtime and let him sleep his last sleep in her bosom: how beardless youths have made haste to inherit their fathers wealth; and how fair damsels- blush not, sweet ones- have dug little graves on the harden, and bidden me, the sole guest to an infants funeral. (Hawthorne) Brown is one of the very people, who among his followers, have been found standing in the path of the Devil, just as those during the Salem Witch Trial. Anyone could have been accused and tried for a crime, only that person knew if they committed. Thus, how could these villagers indict one another when everyone of them were living a sin? Furthermore, the daunting feeling in the forest and the witchery taking place during the story would also represent a sin to the Puritans in Salem and, thus, symbolizes the relationship between oneself and his or her faith. The setting of Salem village for this story is fitting for the ideals Hawthorne is trying to portray to his readers. The fact that all people are capable of cruelty, hatred, and dishonesty- even the church members of the town-shows that anyone may be a friend with the Devil. So, because Salems witches were sinners working for the man himself, it is only fitting that these sinners were also seen at a ceremony with the Devil in the story of Goodman Brown. If the setting was not in Salem but in some other colonial village, the stronghold of faith and a character stepping outside of that faith, may seem peculiar to its readers. Additionally, two of the characters present at the witches sabbath were actually from the Salem Witch Trials (Wesely). Both Goody Cloyse and Martha Carrier would no longer be significant and meaningful in their contribution to the story as a whole. Finally, the dark forest is a clear representation of the moral and psychological turmoil with evil which is where th e women of Salem were bewitched and where Brown was enticed to that night. Without the wooded forest, in both settings, we would no longer feel the unholiness and fear that both Brown felt as well as the community of Salem when this endeavor took place. From this, any other setting may not have led to the connections we can make with Hawthorne and the importance of his past with the telling of his story.

Why Does Investment Spending Depend On Interest Rates

Q: Why does investment spending depend on interest rates, among other factors? A: Investment spending depends on interest rates due to opportunity cost and risk. For example, when interest rates rise, the opportunity cost of your investment also increases. When interest rates are higher investors are willing to pay less for payment in the future. Which in turn leads to a lower rate of investment. The opposite can be said for falls in interest rates that are met will lower opportunity costs. Q: Why would most investments in the economy fail to take place if there were no financial institutions? A: Most investments in the economy would fail to take place if there were no financial institutions because many independent investors do†¦show more content†¦The Federal Reserve is governed by the Federal Open Market Committee (FOMC). This group is comprised of seven members of the Board of Governors and five regional bank presidents that rotate. The Chair has the final say when it comes to monetary policies and decision making which means that professionals across the nation are constantly anxiously waiting for their thoughts and verdicts. (O Sullivan, 283) Q: Identify how the economy makes the transition from the short run to the long run. A: Economies have officially transitioned from short run to long run when the aggregate demand curve intersect the long run aggregate supply curve. When wages and prices are increasing the supply curve will move upward and will continue to do so as long as society continues to produce passed that of potential output (O Sullivan, 201). The economy transitions from short to long run when enough time passes such that no factors are longer fixed. Q: Contrast the pros and cons of protectionist policies. A: Perfectionist policies are defense mechanisms used when economies believe their industries are being threatened by unjust competition in foreign markets. Positive outcomes associated to implementing these policies include defending workers from foreign participants and giving companies the home advantage when trying to buy time to learn how toShow MoreRelatedWhy Does Investment Spending Depend On Interest Rates1106 Words   |  5 PagesQ: Why does investment spending depend on interest rates, among other factors? A: Investment spending depends on interest rates due to opportunity cost and risk. For example, when interest rates rise, the opportunity cost of your investment also increases. When interest rates are higher investors are willing to pay less for payment in the future. Which in turn leads to a lower rate of investment. The opposite can be said for falls in interest rates that are met will lower opportunity costs. Q:Read Moreeccc1266 Words   |  6 Pages(Outside Econ Department) Name: _________________________________ Student ID: _____________________________ Lecture: ________________________________ Multiple Choices    1. Which of the following would NOT be considered part of fixed investment spending (I)? A) Toyota buys a new robot for its automobile assembly line. B) Apple computer builds a new factory. C) Exxon increases its inventories of unsold gasoline. D) An accountant buys a newly built home for herself and her family. E) allRead MoreConsumption Function, Autonomous Consumption, And Marginal Propensity974 Words   |  4 Pagesrelationship between consumption spending and the level of income o C = Ca + by o Example: Income, wealth, interest rates, age, education, and family size are all part of spending, depending on your level of income. o The life-cycle theory assumes that household members choose their current expenditures optimally, taking account of their spending needs and future income over the remainder of their lifetime. †¢ Autonomous Consumption – The part of consumption that does not depend on income o Example: ElectricityRead MoreINTRODUCTION: Keynesian framework IS-LM Model According to Dornbusch (2011), the IS-LM model was1100 Words   |  5 Pagesfor investments and savings in the goods market while LM stands for liquidity preference and money supply in the money market. The model emphasizes the interaction between the goods and assets markets. Spending, interest rates and income are determined jointly by equilibrium in the goods and assets markets. The IS curve shows various combinations of interest rates and levels of income at which the goods markets clear while the LM curve shows that the demand for money depends on the interest rateRead MoreThe Economics of Money, Banking, and Financial Markets1072 Words   |  5 Pagesbetween interest rates on three-month Treasury bills, long-term treasury bonds, and Baa corporate bonds? The interest rate on three-month Treasury bills fluctuates more than the other interest rates and is lower on average. The interest rate on Baa corporate bonds is higher on average than the other interest rates. 2. What effect might a fall in stock prices have on business investment? The lower price for a firm’s shares means that it can raise a smaller amount of funds, and so investment in plantRead MoreThe Economics of Money, Banking, and Financial Markets1056 Words   |  5 Pagesbetween interest rates on three-month Treasury bills, long-term treasury bonds, and Baa corporate bonds? The interest rate on three-month Treasury bills fluctuates more than the other interest rates and is lower on average. The interest rate on Baa corporate bonds is higher on average than the other interest rates. 2. What effect might a fall in stock prices have on business investment? The lower price for a firm’s shares means that it can raise a smaller amount of funds, and so investment in plantRead MoreProject Paper Macroeconomics1161 Words   |  5 PagesWhen prices change, how do we measure real income? When prices change we measure real income with 3. What is unemployment? Why can’t it be driven down to zero? Unemployment is when you don’t have a job. Unemployed people are those who don’t have a job but are actively looking for work. Unemployment cannot be driven down to zero because the lower the unemployment rate is, the harder it will be for businesses to hire new employees. The harder it is to find qualified employees, the more competitiveRead MoreMonetary Policy And Fiscal Policy1178 Words   |  5 Pagesmaintain the market or stabilize the economy during a financial crisis. Monetary policy and fiscal policy are two tools by which government uses to guide the economy. Sometimes the economy is challenged with both inflation and unemployment at high rates. Macroeconomics breaks down the entire economy and the issues affecting it, including inflation, unemployment, economic growth, and monetary and fiscal policy. A country has to come up with good macroeconomic policies in order to better their economyRead MoreThe Study of Macroeconomics1507 Words   |  7 Pagesto the c hanged prices by changing the quantities they make, buy or sell. †¢ Why Macroeconomics Matters†¦ †¢ Cultural Literacy – â€Å"If you want to follow and participate in public debates and discussions, you need to know about macroeconomics.† – Self-interest – â€Å"the macro economy matters to you personally† †¢ Civic Responsibility – â€Å"working together, we can improve the macro economy.† †¢ Macroeconomic Questions †¢ Why do output and employment sometimes fall and how can unemployment be reducedRead MoreU.s. Government Budget Deficit And Debt1351 Words   |  6 Pagesnational and personal debt in the country. First, I am going to start with a definition of U.S government deficit: It is essentially the difference between what the U.S government spends in any fiscal year and the revenues it takes in. Any time the spending exceeds the revenues, the federal government runs a deficit. As many may already know the United States must increase its revenue or reduce expenses during the next decade in order to farther debt and loss of creditability in the world financial

Legalizing Gay Marriage Essay Example For Students

Legalizing Gay Marriage Essay Legalizing Gay Marriage Essay America is the land of the free, which gives each legal citizen the right of privacy, freedom of speech, and freedom of religion. This law also gives each citizen the freedom from religion. If you are Christian, Jewish, Muslim, or Buddhist, you are welcome in this country. However when the topic of gay marriage arises in conversations, the main opposing point would be that the bible doesnt agree with it. According to the first amendment, the bible has no standing in American law. Opposing gay marriage in my opinion is just another form of prejudice that is still being withheld. Im just waiting for someone to explain to me how this argument is any different from the one half a century ago when it was thought almost inconceivable for an African American to share the same water fountain, and ride next to you on a bus. On a common sense level, does American citizens really think that if gay marriage is not allowed, that the gay citizens are going to stop/change their way of life? But of course those who oppose giving citizens their freedom, really have no supporting facts to throw into the argument besides the religious ones. Yet, how do we know for sure that the religious reasons are even factual? Yes, it says so in the bible, whose bible? To my knowledge, as stated above, I wasnt aware that America had a religion that everyone abided to. In fact, most religions are unopposed to gay marriage. Its been said, gay couples are unnatural. Human beings are animal species, and according to scientific studies the 1500 animal species that have been extensively studied, 450 of those animal species have shown homosexual behaviors such as displays of affection, life long bonding, and sex. So just because we are inferior to all other animals on earth doesnt change the fact that our species as a whole has a percent that shows homosexual tendencies. If you ask me, its also unnatural and immorally wrong for a man to beat his wife and children, and many people disagree with it, but it will never be changed. People are going to do what they wish, because every human being is different, and whoever wants them all to be the same, has bigger problems than disagreeing to gay marriage. If a couple, gay or not has a love that deserves a marriage, why deny them that right when others who dont deserve anything in this world are granted it so easily? So what bothers everyone so much? I mean since this is a free country and all, and everyone has privacy and is allowed to do whatever they want behind closed doors, then why cant the bond of two people be made official to the public if they feel the need to do so. Life is too short for people who are happy not to be able to strive for the goals that they really care about. So why waste your time arguing over the happiness of people who arent going to change over a law that the public could make. If youre asking why I feel so strong on this subject, or why i was inspired to write about this, besides the reasons mentioned, I can say that ive lived with gay individuals in my life. One of my best friends in the world, had to come to the realization of being gay last year. So some of those who argue that people that are gay can help it obviously havent met anyone who is. So much of the public is close-minded on this issue because of that fact. My best friend had to come to this realization through people that supported him, and it was hard for him to tell his family. .u0e0c5c30a528a514f05475d7fda34c81 , .u0e0c5c30a528a514f05475d7fda34c81 .postImageUrl , .u0e0c5c30a528a514f05475d7fda34c81 .centered-text-area { min-height: 80px; position: relative; } .u0e0c5c30a528a514f05475d7fda34c81 , .u0e0c5c30a528a514f05475d7fda34c81:hover , .u0e0c5c30a528a514f05475d7fda34c81:visited , .u0e0c5c30a528a514f05475d7fda34c81:active { border:0!important; } .u0e0c5c30a528a514f05475d7fda34c81 .clearfix:after { content: ""; display: table; clear: both; } .u0e0c5c30a528a514f05475d7fda34c81 { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u0e0c5c30a528a514f05475d7fda34c81:active , .u0e0c5c30a528a514f05475d7fda34c81:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u0e0c5c30a528a514f05475d7fda34c81 .centered-text-area { width: 100%; position: relative ; } .u0e0c5c30a528a514f05475d7fda34c81 .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u0e0c5c30a528a514f05475d7fda34c81 .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u0e0c5c30a528a514f05475d7fda34c81 .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u0e0c5c30a528a514f05475d7fda34c81:hover .ctaButton { background-color: #34495E!important; } .u0e0c5c30a528a514f05475d7fda34c81 .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u0e0c5c30a528a514f05475d7fda34c81 .u0e0c5c30a528a514f05475d7fda34c81-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u0e0c5c30a528a514f05475d7fda34c81:after { content: ""; display: block; clear: both; } READ: Great Gatsby Essay Everyone on this earth is different, and I think that we as people should value that. Just let them enjoy their lives, because even if they arent exactly like you or have the same sexual preference, they are human beings and deserve the rights of one. .

Queen Elizabeth Grammar School Essay Example

Queen Elizabeth Grammar School Essay The Towers of Hanoi is an ancient mathematical game. The aim of this coursework is to try to identify patterns and rules associated with the game and explain them in mathematical terms. The definitions and rules are: Rules: * There are only three positions a disc can be placed. Poles A, B or C. * A disc can only go on top of a larger one. (I.e. Disc A can only go on top of Discs B and C, but Disc B cannot go on top of disc A) We will write a custom essay sample on Queen Elizabeth Grammar School specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on Queen Elizabeth Grammar School specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on Queen Elizabeth Grammar School specifically for you FOR ONLY $16.38 $13.9/page Hire Writer * The object of the game is to get all the discs to move from pole A to pole B of C in the least number of moves. * Only one disc may be moved at a time. Finding Formula A Number Of Discs Least Number Of Moves Previous term (Doubled) 1 1 2 3 2 3 7 6 4 15 14 5 31 30 6 63 62 7 127 126 8 255 254 From looking at the table it is quite clear that there is a pattern linking the number of discs and the least number of moves. It is clear that there is an element of doubling involved, as the least number of moves nearly doubles each time. When I add the extra column see above, it is clear that there is a doubling element involved. When I look again, I can see that the pattern is the previous term doubled plus 1. This can be expressed mathematically as: Un = 2(Un-1) +1 This can be shown in: 1. For 1 disc, it takes 1 move to move disc A from pole 1 to pole 3; 2. For 2 discs, it takes 3 moves: 2(Un-1) +1 = 2(1) + 1 = 3 3. For 3 discs, it takes 7 moves: 2(Un-1) +1= 2(3) + 1 = 7 4. For 4 discs, it takes 15 moves: 2(Un-1) +1= 2(7) + 1 = 15 5. For 5 discs, it takes 31 moves: 2(Un-1) +1= 2(15) + 1 = 31 To understand how this works, coding is needed to see how a disc moves individually. Coding should show me the patterns involved and then I should be able to justify my formula based on this. Coding is on the next page. Coding Number of Discs: 2 3 4 5 Disc Moving: A A A A B B B B A A A A C C C A A A B B B A A A D D A A B B A A C C A A B B A A E A B A C A B A D A B A C A B A From my coding it is now clearer why that formula is that particular formula. It can be seen that there is symmetry involved in each pattern. The symmetry is always about the name of the bottom disc. I.e. with 3 discs the symmetry is about disc C and this is the bottom disc From the coding, I can also see that the pattern of moves for 2 discs is present in the beginning of 3 discs, 4 discs etc. The pattern for 3 discs is also in the pattern for 4 discs and so on. This is can therefore be explained as: In n number of discs where n is greater than 2, the first three moves will always be ABA. This is because the n-1 discs pattern is included in the n pattern. We have (Un-1), because we take into account the previous terms pattern when making the next tower. We have the 2 term because this pattern is repeated twice, firstly to deconstruct the tower and then to rebuild the tower on top of the bottom disc. We have the +1 term because this is where the bottom disc moves from Pole A to Poles B or C. This can be demonstrated when we move three tiles. ABA This is the move pattern for 2 tiles (Un-1). This allows C to be able to move. C This is when the bottom tile moves and we therefore get the +1 from. ABA This is where the doubling element comes in as well as the n-1 discs moves pattern. This is where the tower is rebuilt on top of disc C. So overall, we get the formula: 2(Un-1) +1 There are limitations to this however. Un-1 has to be an integer because we cannot have 3.5 moves. Un-1 has to also be equal to or greater than 0 and has to be an integer because the formula wouldnt work as the result would be negative and we cannot have a negative number of moves. Formula B Finding the formula that shows how many times a certain disc moves From formula A I now have a basis on which to work. Given a certain number of discs I need to be able to say how many times a desired disc moves. Firstly, I need to analyze my results from the coding. Disc: Disc A Disc B Disc C Disc D Disc E Disc F Total Number of times each disc moves: 3 Discs 4 2 1 7 4 Discs 8 4 2 1 15 5 Discs 16 8 4 2 1 31 6 Discs 32 16 8 4 2 1 63 We can also once again see a pattern here. There is a doubling, well halving element involved depending on which way you look at it. The table above shows how many times a certain disc moves. Whenever a new disc is added to the sequence, such as in Disc 4, the number of moves for Disc A doubles. I.e. As you go down the table the number of moves for each disc doubles. When I look at the results, I notice that they are all from the 2n pattern. Therefore I can come up with the relationship for the number of times each disc moving being: Number of times a certain disc moves = 2n-d with d being the disc number. So in Disc A, the number for d would be 1, as this is the first disc. Disc B would be 2 etc. In the series for 6 discs, the terms would be Disc A: 2n-d = 26-1 = 32 Disc B: 2n-d = 26-2 = 16 Disc C: 2n-d = 26-3 = 8 Disc D: 2n-d = 26-4 = 4 Disc E: 2n-d = 26-5 = 2 Disc F: 2n-d = 26-6 = 1 This therefore works. Now I have to prove that this works. We can see that Disc B always moves half as many times as Disc A. If we do 2n we get how many times Disc A Moves always. If we do 2n-2 we get how many times disc B moves always. This is because as we take more away from 2n we get smaller and smaller until it ultimately converges to 0. Taking 1 away from this halves the number of moves; whereas taking 2 away quarters the number of moves. Disc B always moves less times than Disc A because of the recurring pattern. A has to move more times, because it has to keep going on top of the larger tiles as the rules state. A has more options to move than B because it is smaller. There are limitations to this however, because we cannot have d being greater than n because the formula would not work. It wouldnt work because we cannot have half of a move or a quarter of a move. We cannot also have n being less than 1 because of the same principal. The number of moves and the disc number have to also be an integer because we cannot have Disc A moving 3.5 times. The Link The series above is a geometric series. I know this because the difference is different each time. The general way to write a geometric series is: General: a + ar + ar2 + ar3 + arn-1 The terms: a is the starting number in the sequence. I will use a 6 tiled sequence so my starting number from the table will be 32 as this is the number of times disc A moves. Ratio r This is the amount that a is multiplied to get the next term. So 32 is multiplied by 0.5 to get 16. Our sequence is: S= 32 + 16 + 8 4 +2 +1 To get the sum of a geometric sequence, we need to multiply by the common ratio (0.5) S = a + ar + ar2 + ar3 + + arn-1 rS = a + ar + ar2 + ar3 + +arn-1+ arn S-rS = a arn This can be expressed as S(1-r) = a(1-rn) Divide this by 1-r gives: a(1-rn) S= 1-r Before I can use this information however, I need to determine a formula to get a. I can use the formula I discovered above but just modify it slightly. To get a the formula is: 2n-1 as this is the formula for Disc A always. So the formula above instead of being 2n-d could have also been 2n-1 for the same principals. With n being the disc number you are trying to find. Disc 50 would be 249 and disc 3 would always be 22 and so on. Therefore I can now substitute in my values in a pile of 6 discs to get the formula that links formula A and B. To determine the ratio we have to just see how much the sequence is decreasing each time. 32 + 16+ 8 + 4 + 2 + 1 To get the next term suing the general geometric sequence rule, it says that we have to multiply 32 by a constant. a ar. So: a ar is the same as ar divided by a. 16 = 0.5. This is the ratio. 32 Therefore for the sum of my geometric series, the formula should be: a(1-rn) S= 1-r S = 25(1-0.56) 0.5 S= 31.5 = 63 0.5 Therefore the sum of a 6 termed series is 63. This can be proved by getting the formula for the previous term. S = 24(1-0.55) = 31 0.5 According to my earlier formula (Un = 2(Un-1) +1) when I substitute in I should get the answer 63. 2 x 31 + 1 = 63. This works because of the algebra of the general geometric sequences: S = a + ar +ar2 This is the rule for each term in the sequence: arn-1 or 2n-d I then multiplied by the common ratio (r) rS = ar + ar2 + ar3 This is the rule: arn Then I subtracted the sequence multiplied by the common ratio from the first sequence. This gave: S-rS = a arn = a(1-rn) Therefore S=a(1-rn) (1-r)) Limitations are: S has to be greater than 0 and has to be an integer a has to be positive and an integer r has to be an integer and greater than 0 Extension Work: Finding which pole the pile will be built upon. I have noticed from my work that when I had 3 discs on my pile, disc C landed on where I put disc A to start off with which was on Pole C. When I had 4 discs however, I noticed that the pile finished on where I did not place tile A which was Pole B. This can therefore be expressed as: If the number of discs in the pile to start with is even then the bottom disc will land where you place Disc A to start off with. If the number of discs in the pile is odd however, then the bottom disc in the pile will finish up on the pole where you did not place Disk A. Therefore where you put Disc A can be considered crucial to where you want your pile to land Overview: If I have 25 discs in my pile, I can expect there to be: 33554431 moves involved in the series. Disc A will move 16777216 times; whereas disc y will move only once. The Pile will end up on the pole where you place disc A, so if I leave it on pole B to start with, the pile will end up on Pole B. According to the monks in Hanoi, the world will end in over 500 Million Years. The problems with my investigation: I have realized that there are only 26 letters in the alphabet. With my system of labeling, it is impractical for me to label each disc A, B C etc because I will run out of letters. I will either have to name the poles ABC or call each disc past 26, A1 etc.