Behavioural Issues Within Budgeting Essay

Objectives of Budgeting Through budgeting organisations can provide information for strategic planning and control, these are the two main objectives of the budgetary control system. Management and management accountants must work together and operate a system that achieves these objectives, they do so through a system called variance analysis. Management accountants compare the actual results against the budgets; they then send reports to the management concerning the extent to which budgets are being met. Management can then control activities by making possible steps to stop situations where the budget is being ignored or overlooked. To meet their controlling and planning needs, management and management accountants adapt the feedback and feed-forward principles (types of controls). As well as feedback, budgetary systems consist of feed-forward controls, where expected outcomes can be compared with desired outcomes. A recent report Tayles (1998) suggests that, â€Å"†¦feedforward control consists of a prediction being made of anticipated future outputs. If the expected outputs differ from what outputs are desired, control actions are implemented to minimise these differences. Control is therefore, achieved, if the control actions are effective, before any deviation from the objective output occurs†. Feedback is the detection of a deviation between actual results and an objective; normally this is carried out after the event and is essentially error based. Research has shown that up to date, accurate feedback has a motivational effect, delay and inaccurate data are demotive. Most organisations adopt th ese controls, as they are effective and aid the pursuit of a budgetary control system. Target setting is another objective of budgeting and may possibly have motivational benefits. Setting targets that are realistic and clearly stated will encourage employees to make more effort to achieve it than they might otherwise do. However, the motivational effect of budgetary control is far from clear, as we shall see later in this report. In addition, there is no  point in having targets and controls if they are kept a secret. Through communication, organisations can enhance the level of information that flows from top management to employees (subordinates). These are just some of the arguments for budgeting. The rest of this report will concentrate on the behavioural issues in a budgetary control system. Behavioural Considerations When drafting and planning a budgeting system there are behavioural issues to consider as budgets can have implications for human behaviour and, budgeting can have harmful side effects on performance. There is a danger that employees will concentrate solely on the objectives set by the budget, regardless of whether or not it is organizationally desirable. This means that individuals would set out to exploit the rules of the budgetary control system to boost their individual performance, ignoring other areas not monitored by the control system. Drury (2000, p.600) found that â€Å"†¦actual behaviour may be modified so that desired results appear to be obtained, although they may have been achieved in an undesirable manner which is detrimental to the firm†. In addition, budgets may give management a means of judging the performance of their teams. It’s assumed that, the setting of performance measures should complement the setting of new strategies and objectives and the implementation of action plans for the organisation. However, that adherence to the budget alone cannot measure all aspects of a manager’s performance. Therefore, it is important that managers achieve their targets frequently enough to give positive feedback in their efforts. Another behavioural aspect to consider when introducing a budgetary control system is the controls set by the new system, as these may cause negative attitudes. A potential cause of negative attitudes is the way a control is applied, if targets are considered unachievable and applied too rigorously, they may cause negative attitudes. This may also lead to harmful side effects such as the lack of goal congruence and organisational performance. However, if care is taken in designing the control system negative attitudes may be avoided. Drury (2000, p.601) makes a similar argument, â€Å"The way that  a control system is applied can be just as important as the design issues in determining the success of a control system†. Importance of Behavioural Considerations in the Budgetary Process Participation Interaction of managers and employees or budgetees to the targets that are incorporated in their budgets is known as participation. Tayles (1998) suggests that, â€Å"†¦participation in the budget process and discussion over how results are to be measured has benefits in terms of budget attitude and performance of the budgetee†. This would be of an advantage to an organisation as employees would be more receptive of the targets and more committed to achieving them. In addition, the levels of staff morale would be greater than before. Participation can also improve communications and tends to improve the degree to which budgets are met. For example, participation creates a common understanding of the organisations objectives and makes achieving goals more likely. Communication Communication is a necessary activity in all factors of management and can be broadly defined as an exchange of information to bring about a mutual understanding between two parties. Welsch (1988) Defining or clarifying the lines of communication within an organisation can be a powerful aid in the construction of the budget. Communication can have an important part to play in the budgets objectives, targets and responsibilities throughout the organisation. Carried out properly, this can have considerable benefits in promoting co-operation at all levels. Therefore, in order for an organisation to wok effectively there must be lines of communication so that all parts can de kept fully informed of the part they are expected to play in achieving the budget. This aspect can have important behavioural implications throughout an organisation as the attitude of the person who receives the communication may be affected. Motivation By setting challenging but realistic targets, well-designed budgets can play a significant part in motivating managers and employees to perform in line with the organisational objectives. The targets must be clear and achievable, and the manager should participate in setting his or her own budget, as it is more likely to be acknowledged. Generally, it’s believed for budgets to motivate, higher objectives should be set. The levels of expected performance that are set influence the motivation of managers responsible for target achievement. If levels are set too high, then there is a strong disincentive to management involvement in the budgetary process, and a low level of motivation. It levels are set too low, then managers can achieve targets easily despite inefficiencies. This is known as budgetary slack. Setting appropriate levels of attainment in budgets is a complex and difficult activity with an important behavioural dimension. Therefore, it can be said that motivation is a process of arousing and sustaining goal-directed behaviour induced by the expectation of satisfying individual needs. Goal Congruence Goal congruence means that as people work to achieve their own goals, they also work to achieve the goals of the organisation. Nevertheless, there is a danger that employees will concentrate only on what is been monitored, leading to a lack of goal congruence. Focused on preventing this undesirable behaviour and encouraging goal congruence is controls, for example, action or behavioural controls. Action or behavioural controls involves managers watching employees as they go about there work and if the managers know what actions are desirable (or undesirable) the desired outcomes will come about. Performance measures may be used as a good indicator of what is desirable to achieve an organisations goals and some measure may encourage goal congruence or organisationally desirable behaviour. For this to be effective management must also ensure  that desired actions are taken. Goal Definition A further use of budgets is as a basis for setting performance standards and rewards, for example, bonus, status or enhanced promotion prospects are often linked to budget attainment. This may be defined as a process theory of motivation that focuses on the process of setting goals, Emmanual (1992). It is argued that the natural human preference to set and strive for these goals is useful only if the individual both understands and accepts a particular goal. Therefore fundamental to goal attainment is: *an understanding and acceptance of a particular goal (goal congruence) *skills to achieve the goal *confidence that they have the skills to achieve the goal *a reasonable expectation of achieving that goal People work more efficiently when they have goals and targets. Therefore, if the targets are agreed and accepted by individuals, they should achieve goal congruence and motivation. Conclusion It will be apparent from this report that budgets serve various purposes and in some cases, these purposes can be in conflict and have a consequent effect on management and employee behaviour. Nevertheless, if controls are put in place that focuses on preventing undesirable behaviour, employees of an organisation would be discouraged from taking part in such actions. Budgets serve as a means of forecasting, planning, control and a channel of  communication and motivation. But, a good budgetary system should be designed to: *meet the objectives regarded as most important by senior management *provide information which is useful in meeting those objectives *Reduce the risk of unintended and undesirable behaviours. REFERENCES Drury, C. (2000) Management & Cost Accounting, 5th edition, Thomson Learning Emmanual, C. Otley, D. Merchant, K. (1992) Readings in Accounting for Management Control, Chapman & Hall Tayles, M. (01 Dec 1998) budgetary control – the organisational aspects Available at: www.acca.com [Accessed 7 April 2002] Welsch, G.A. Hilton, R.W. Gordon, P.N. (1988) Budgeting – Profit Planning and Control, 5th edition, Prentice Hall

Aims of Germany and Japan essays

Aims of Germany and Japan essays The 1930s were a tumultuous time. After World War I, the Allied powers seemed determined to preserve peace, but Germany and Japan held a shared goal of world domination. In two memorandums about the plans of Germany and Japan, U.S. officials make the position that the United States should take plain: mobilize and be ready for war, but do not provoke it. In other words, Speak softly but carry a big stick. Although Germany and Japan had a few similar goals and ideas, they differed on other issues of subject peoples and races. Both Germany and Japan had plans to rule the world. They began their conquests in the early- to mid-1930s, Japan with Manchuria in 1931 and Germany with the Rhineland in 1936. These occupations would eventually lead to WWII. Germany and Japan also wanted to control trade and have a political influence in the countries that surrounded them. For Germany, it was a matter of national pride. They wanted to show that Germany was still a great state after their humiliating defeat in WWI. The expansionists of Japan felt that Japans destiny is to subjugate and rule the world. There was no tinge of revenge to the The main difference between Germany and Japan was that the Japanese military had the power to defeat other countries without the permission of the government. The German government not only gave permission, they persuaded the German people to support their hunger for territory. Germany was intensely xenophobic. All peoples other than Aryans were persecuted. They intended to annihilate the Jews and breed out the Slavic people. The memorandum concerning Japan does not give evidence of xenophobia in Japan, but neither does it imply that the Japanese were Germany and Japan had many similarities in their foreign policies. The ...

Chicago Referencing †Repeat Citations - Proofeds Writing Tips Blog

Chicago Referencing – Repeat Citations Chicago Referencing – Repeat Citations If you have a useful source text, you’ll often want to cite it more than once in your work. But the Chicago Manual of Style has specific rules for doing this, so you need to how it works! Here, then, is our guide to repeat citations in Chicago style referencing. Consecutive Repeat Citations In Chicago footnote referencing, when you cite the same source twice in a row, you can use the Latin abbreviation â€Å"ibid.† This literally translates as â€Å"in the same place.† If you are citing exactly the same page as before, you can use â€Å"ibid.† by itself. If you are citing a different part of the text, you should give the new page number(s), too. For instance: 1. Alan C. Jenkins, Wildlife in the City: Animals, Birds, Reptiles, Insects and Plants in an Urban Landscape (London: Holt Company, 1983), 13. 2. Ibid. 3. Ibid., 102. Here, page 13 of Alan C. Jenkins’ Wildlife in the City is cited in footnotes 1 and 2, while page 102 is cited in footnote 3. This saves us from writing out the full source information each time. Non-Consecutive Repeat Citations What, though, do you do when you need to repeat a citation after you’ve cited another source? In this case, Chicago referencing uses a shortened footnote style. These shortened footnotes should include: The author’s surname The page(s) cited Non-consecutive citations of the same source would therefore look like this: 1. Alan C. Jenkins, Wildlife in the City: Animals, Birds, Reptiles, Insects and Plants in an Urban Landscape (London: Holt Company, 1983), 13. 2. Esther Woolfson, Corvus: A Life with Birds (London: Granta Publications, 2008), 234. 3. Jenkins, 102. If you have more than one source by the same author, however, you should also include a shortened version of the title. This will ensure the reader knows which source you’re citing. For instance: 1. Alan C. Jenkins, Wildlife in the City: Animals, Birds, Reptiles, Insects and Plants in an Urban Landscape (London: Holt Company, 1983), 13. 2. Esther Woolfson, Corvus: A Life with Birds (London: Granta Publications, 2008), 234. 3. Alan C. Jenkins, Introducing Cats (London: Spring Books, 1958), 24. 4. Jenkins, Wildlife in the City, 102. In this case, footnotes 1 and 4 point to the same source, while footnote 3 is another source by Jenkins. Author–Date Citations Chicago referencing also has an author–date system, which uses in-text citations. To reference the same source more than once in this, all you have to do is give the same citation again: Alan Jenkins (1983) describes how birds of prey survive in urban settings. He says that peregrine falcons are a â€Å"spectacular example of adaptive behavior† (Jenkins 1983, 13). All you need to do with repeat author–date citations, then, is make sure they are consistent!

Theories about whether or not virtue can be taught Essay - 93

Theories about whether or not virtue can be taught - Essay Example In the Meno, Socrates states that virtuous and wise people are not very good at imparting these qualities to others. For example, while Themistocles, the Athenian statesman and soldier could teach the skills of shooting javelins and standing upright to his son Cleophantus, no one ever stated about Cleophantus that he was virtuous and wise, and this holds true for Pericles and his sons Xanthippus and Paralus; Lysimachus and Aristides, his son; and Thucydides and his sons Stephanus and Melesias. Since there are apparently no teachers of virtue, it was inferred by Socrates that it is not possible to teach virtue; and since it cannot be taught, virtue cannot be translated as a kind of knowledge. Meno questions that if it is not possible to teach virtue, then how could good men exist? Socrates retorts that Meno and he overlooked that it is possible to execute a right action without knowledge. This can be explained in terms of the quality of guidance offered by two men to Larisa, one of wh om has been there whereas the other just knows about the way and has never actually been to Larisa. Of the two, the one who knows the way to Larisa would serve as a good guide, but so would be the other man who has never been to Larisa. If it is believed that truth as a guide is just as good as the man who is aware of that truth, it implies that a correct opinion is no less good a guide for the correct action than knowledge. In this case, Meno wonders what the factors that distinguish knowledge from correct opinion are, and also asks why should knowledge be preferred over correct opinion or vice versa? Socrates addresses these questions by saying that correct opinions can be interpreted as Daedalus’s statues, who was a highly skilled craftsman and mythological architect. The statues of Daedalus needed to be tied to prevent them from running away. Similarly, it is possible to tie down the correct opinions with the reason whereupon they discontinue being correct opinions and  becoming a form of knowledge.  

Theory evolution Research Paper Example | Topics and Well Written Essays - 250 words

Theory evolution - Research Paper Example They emerged with a goal of strengthening the already existing methods. The original ideas had some weaknesses in their area of scope, field of coverage and failed to pinpoint in some of the most sensitive aspects that indeed can improve the process of understanding the adult theory. The alternative methods added some knowledge on adult learning, like, for example; they came up with other principles of adult education that got neglected by the original theories. These included the assumptions of self-concept, adult education experience, readiness to learn, orientation to learn and motivation to learn. These new assumptions that got added to the alternative theories brought an insight and knowledge on how to approach adult education (Jarvis & Jarvis, 2004). The alternative methods contributed other principles of Andragogy. The new laws gave other means through which the process of adult learning can get well comprehended (Flannery, 1993). These methods have brought in some insights towards understanding of the adult education. Scholars have gotten in a position to get awareness on the flexibility and dynamism that Andragogy exhibits over time (Merriam, 2008). Andragogy has further gotten understood as a process that has extreme differences in pedagogy, due to the different assumptions and principles that are parallel to those of

Different ways to compete in the soft drink business Case Study

Different ways to compete in the soft drink business - Case Study Example Through the sales of internet routers, the company managed to raise high revenue of approximately ten billion dollars. However, this was brought to an end in the year 2008 when the demand of products from the company such as internet hardware dropped. At this point, the best action to be taken by any manager who wants to save the company from collapsing is to make adjustment on the business plan and strategies so as to fix the problem. John Chambers was very wise and keen on the issue; after realizing that the market of internet router could not raise the great profits that were required to match the standards of the company, he embarked on another business plan. The plan was to shift in to very new markets and industries. One of the strategies was to expand in a rapid way from the software business and the core internet hardware to the consumer electronic industry. His first step to implement the plan was to associate with companies that were already producing products related to th e internet. The CEO sought to get hold of the companies that dealt with internet bandwidth products so as to increase the consumer demand since this would result in to a high demand for the Cisco`s products. Through John Chambers, Cisco made an announcement in the year 2009 that it would make a payment of $590 for purchasing pure digital for making video camcorders that are more colorful and are pocket sized. This further increased the financial status of Cisco due to the fact that over two million people could access and share the video through the web (Jones & George, 263-264). If I were the manager of Cisco system, I could have done the same things he did or come up with a plan with strategy for implementation that would end up in high performance. It is always believed that better plans always reflects on a bright future for the business and that lack of a business plan results in to hesitations , mistaken business decisions and false steps. Having this in mind, I would make a p lan for a good number of reasons. It`s importance is to give the company proper direction for the right purpose since the plan contains the objectives that the organization has to achieve and the strategies to be used. I believe this would have a great impact on the company so that as soon as the demand of the Internet s products provided by the Cisco Company goes down, then the strategies of the plan would help save the company from collapsing. This could be done by linking the company to other industries that produce items that are the same as the ones produced by Cisco to come up with a better opinion to help increase the consumer demand (Covey, 162). The challenge of proper planning and formulating a proper strategy to implement it was well overcomed by the CEO of Samsung electronics, Lee Kun. In the year 2000, the company which was mainly based in Korea emerged to be in the second position in terms of profit among the companies in global technology coming just after the Microso ft. There were strategies and steps that were followed by the organization which enabled them to attain their targeted goal. Under the guidance of Lee Kun, Samsung electronics developed and built competencies as the first step in the manufacturing of low-priced products. The second step was to put all the efforts in the manufacture of R&D. At the end, the organization used those

Applications Of The Pigeonhole Principle Mathematics Essay

Applications Of The Pigeonhole Principle Mathematics Essay We begin our discussion with a common daily embarrassing moment. Suppose that in ones dresser drawer, he has socks of three different colours (all placed in messy order). Having to get up early in the morning while it is still dark, how does he ensure that he gets a matching pair of same coloured socks in the most convenient way without disturbing his partner? While, the answer is simple! He just has to take 4 socks from the drawer! The answer behind this is of course, the Pigeonhole Principle which we will be exploring in this Maths Project. What is the Pigeonhole Principle then? Let me give you an example to illustrate this principle. For instance, there are 3 pigeonholes around. A pigeon is delivering 4 mails and has to place all its mails into the available pigeonholes. With only 3 pigeonholes around, there bound to be 1 pigeonhole with at least 2 mails!. Thus, the general rule states when there are k pigeonholes and there are k+1 mails, then they will be 1 pigeonhole with at least 2 mails. A more advanced version of the principle will be the following: If mn + 1 pigeons are placed in n pigeonholes, then there will be at least one pigeonhole with m + 1 or more pigeons in it. The Pigeonhole Principle sounds trivial but its uses are deceiving astonishing! Thus, in our project, we aim to learn and explore more about the Pigeonhole Principle and illustrate its numerous interesting applications in our daily life. We begin with the following simple example: 2. Pigeonhole Principle and the Birthday problem We have always heard of people saying that in a large group of people, it is not difficult to find two persons with their birthday on the same month. For instance, 13 people are involved in a survey to determine the month of their birthday. As we all know, there are 12 months in a year, thus, even if the first 12 people have their birthday from the month of January to the month of December, the 13th person has to have his birthday in any of the month of January to December as well. Thus, we are right to say that there are at least 2 people who have their birthday falling in the same month. In fact, we can view the problem as there are 12 pigeonholes (months of the year) with 13 pigeons (the 13 persons). Of course, by the Pigeonhole Principle, there will be at least one pigeonhole with 2 or more pigeons! Heres another example of the application of Pigeonhole Principle with peoples relationship: 3. Pigeonhole Principle and problems on relations Assume that the relation `to be acquainted with is symmetric: if Peter is acquainted with Paul, then Paul is acquainted with Peter. Suppose that there are 50 people in the room. Some of them are acquainted with each other, while some not. Then we can show that there are two persons in the room who have equal numbers of acquaintances. Lets assume that there is one person in the room that has no acquaintance at all, then the others in the room will have either 1, 2, 3, 4, à ¢Ã¢â€š ¬Ã‚ ¦, 48 acquaintance, or do not have acquaintance at all. Therefore we have 49 pigeonholes numbered 0, 1, 2, 3, à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦, 48 and we have to distribute between them 50 pigeons. So, there are at least two persons that have the same number of acquaintance with the others. Next, if everyone in the room has at least one acquaintance, we will still have 49 pigeonholes numbered 1, 2, 3, à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦, 48, 49 and we have to be distribute between them 50 pigeons! Also, we can apply the Pigeonhole Principle in the proving of numerical properties. The following are two of such examples: 4. Pigeonhole Principle and divisibility Consider the following random list of 12 numbers say, 2, 4, 6, 8, 11, 15, 23, 34, 55, 67, 78 and 83. Is it possible to choose two of them such that their difference is divisible by 11? Can we provide an answer to the problem by applying the Pigeonhole Principle? There are 11 possible remainders when a number is divided by 11: 0, 1, 2, 3, à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦.., 9, 10. But we have 12 numbers. If we take the remainders for pigeonholes and the numbers for pigeons, then by the Pigeon-Hole Principle, there are at least two pigeons sharing the same hole, ie two numbers with the same remainder. The difference of these two numbers is thus divisible by 11! In fact, in our example, there are several answers as the two numbers whose difference is divisible by 11 could be 4 15; 34 67 or 6 83. 5. Pigeonhole Principle and numerical property We can also apply the Pigeonhole Principle in determining useful numerical properties. Consider a sequence of any 7 distinct real numbers. Is it possible to select two of them say x and y, which satisfy the inequality that 0 The problem sounds difficult as we may need to consider more advanced calculus and trigonometrical methods in the determination of the result. Well, to answer the above problem, one will be surprised to know that we just need a simple trigonometrical identity and apply the Pigeonhole Principle! Before proceeding to answer the problem, we first note that given any real number x, we can always find a real number a where n1 = tan a1, n2 = tan a2, à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦.., n7 = tan a7 Now, if we were to divide the interval (-p, p) into 6 equal intervals, we obtain the following sub-intervals: ( -p, -p ), [ -p, -p ), [ -p, 0 ) , [ 0, p ), [p,p ) and [p, p ). For the 7 distinct numbers a1, a2, à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦.a7, by the Pigeonhole Principle, there should be two values say, ai and aj such that aI > aj and ai aj are in the same interval! For these two values ai and aj, we should have 0 We may recall an important trigonometrical identity: B ) = . Thus, if ni = ai and nj = aj , then = = tan ( ai aj ) As 0 0 and so, 0 which is the result we are seeking! We may also apply the Pigeonhole Principle in the proving of useful daily geometrical results.. The following examples illustrate such usages: 6. Pigeonhole Principle and Geometry a. Dartboard applications Another common type of problem requiring the pigeonhole principle to solve are those which involve the dartboard. In such questions, a given number of darts are thrown onto a dartboard, the general shape and size of which are known. Possible maximum distance between two certain darts is then to be determined. As with most questions involving the pigeonhole principle, the hardest part is to identify the pigeons and pigeonholes. Example 1: Seven darts are thrown onto a circular dartboard of radius 10 units. Can we show that there will always be two darts which are at most 10 units apart? To prove that the final statement is always true, we first divide the circle into six equal sectors as shown; Allowing each sector to be a pigeonhole and each dart to be a pigeon, we have seven pigeons to go into six pigeonholes. By pigeonhole principle, there is at least one sector containing a minimum of two darts. Since the greatest distance between two points lying in a sector is 10 units, the statement is proven to be true in any case. In fact, it is also possible to prove the scenario with only six darts. In such a case, the circle is this time divided into five sectors and all else follows. However, take note that this is not always true anymore with only five darts or less. Example 2: Nineteen darts are thrown onto a dartboard which is shaped as a regular hexagon with side length of 1 unit. Can we prove that there are two darts within units of each other ? Again, we identify our pigeonholes by dividing the hexagon into six equilateral triangles as shown below. With the six triangles as our pigeonholes and the 19 darts as pigeons, we find that there must be at least one triangle with a minimum of 4 darts in it. Now, considering the best case scenario, we will have to try an equilateral triangle of side 1 unit with 4 points inside. If we try to put the points as far apart from each other as possible, we will end up assigning each of the first three points to the vertices of the triangle. The last point will then be at the exact centre of the triangle. As we know that the distance from the centre of the triangle to each vertex is two-third of the altitude of this triangle, that is, units, we can see that it is definitely possible to find two darts which are units apart within the equilateral triangle! b. Encompassing problems Consider the following problem: 51 points are placed, in a random way, into a square of side 1 unit. Can we prove that 3 of these points can be covered by a circle of radius units ? To prove the result, we may divide the square into 25 equal smaller squares of side units each. Then by the Pigeonhole Principle, at least one of these small squares (so call pigeonholes) should contain at least 3 points (ie the pigeons). Otherwise, each of the small squares will contain 2 or less points which will then mean that the total number of points will be less than 50 , which is a contradiction to the fact that we have 51 points in the first case ! Now the circle circumvented around the particular square with the three points inside should have radius = = = It will be worthwhile to note the above technique can be useful in analyzing accuracy of weapons in shooting practices and tests. Next, we will like to proceed to a more creative aspect of the application of Pigeonhole Principle by showing how it can be used to design interesting games: 7. Application of pigeonhole principle in card games We like to introduce the application of pigeonhole principle in two exciting card tricks: a. Combinatorial Card Trick : Heres the trick: A magician asks an unsuspecting observer to randomly choose five cards from a standard deck of playing cards. The participant does not show these cards to the magician, but does show them to the magicians accomplice. The accomplice looks at the five cards, chooses four of them, and shows these four to the magician in a certain ordered manner. The magician immediately identifies the fifth hidden card. How does the trick work? The following is an explanation of our working strategy: (1) First of all, notice that in any hand of five cards there must be two cards of the same suit (an application of Pigeonhole Principle). The first card that the accomplice shows to the magician is one of these two cards. The other card of the same suit is never shown it is the mystery card, the card which the magician must discover. Thus, the accomplice can easily communicate the suit of the hidden card: the hidden card has same suit as the first card shown to the magician. Specifying the rank of the mystery card (ie its value) is a little trickier but can be accomplished with a little circular counting manner which we will explained below Number the cards in a suit circularly from 1(ace) to 11 (jack), 12 (queen) and 13 (king) so that 1 follows 13 i.e. the list is ordered in a clockwise direction. Now, given any two cards A and B, define distance (A,B) as the clockwise distance from A to B. It is easy to see that for any two cards A and B either distance(A,B) or distance(B,A) must always be less than or equal to 6. Again as an application of the Pigeonhole Principle, we note that if they are both 7 or more, then there will be at least 2 x 7 = 14 cards in a standard suit of cards!! Example Cards: 3 and Jack (11) distance(Jack, 3) = 5; distance (3, Jack) = 8 Cards: Ace(1) and 7 distance (Ace, 7) = 6; distance (7,Ace) = 7 (2) Our working strategy thus proceeds as follows.: From those two cards of the same suit, A and B, the accomplice shows the magician card A such that distance(A, B) is 6 or less. For example, given the choice between the three of clubs and the Jack of clubs, the accomplice reveals the Jack (since distance (Jack ,3) = 5 and distance(3, Jack)= 8). The three of clubs remains hidden. If the two same-suit cards are the five of hearts and the six of hearts, the accomplice chooses the five (since distance (5,6) = 1 but distance (6,5) = 12) leaving the six of hearts as the mystery card. (3) Finally, the accomplice arranges the last three cards to encode a number from 1 to 6 the distance from the value of first card to that of the hidden card. A quick calculation allows the magician to discover the value of the mystery card. Notice that although the magician must decode only one of 6 possibilities, it should not present a problem, even to the slowest of magicians. To facilitate the explanation for the last step involved, we may assign each card a number from 1 to 52 for ranking purpose. For example, the ace of spade can be numbered 1 (the highest ranking card), ace of heart numbered 2, ace of club numbered 3, ace of diamond numbered 4, king of spade numbered 5, à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦.., queen of spade numbered 9, à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦., jack of spade numbered 13, à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦., 10 of spade numbered 17, à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦. , à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦. , 2 of diamond numbered 52 (the lowest ranking card). We will now proceed to explain the last step using the following example: Example: Suppose the five cards chosen are the following: 3 of Hearts (numbered 46) 5 of Spades (numbered 37) 6 of Clubs (numbered 35) 7 of Hearts (numbered 30) 2 of Diamonds (numbered 52) The accomplice notices that the 3 and the 7 have the same suit hearts. Since the distance( 3 ,7) = 4 and distance(7, 3) = 9, the accomplice chooses the 3 as the first card to show the magician, leaving the 7 of hearts as the hidden card. The magician now knows that the suit of the mystery card is hearts. The accomplices next task is thus to let the magician know that he must add the value 4 to the number 3 to obtain the final value of 7 for the hidden card! How can he achieve this? Basically, he can arrange the other three cards in 3! = 6 ways. Based on the numbering method explained earlier, the 3 remaining cards can be ranked 1st, 2nd and 3rd . In our example, the 6 of Clubs will be ranked 1, the 5 of Spades will be ranked 2 and the 2 of Diamonds will be ranked 3. The accomplice may agree with the magician earlier that the arrangement of these 3 cards represent specific numbers as shown below: Order in which 3 remaining cards are shown Number represented by the arrangement 1, 2, 3 1 1, 3, 2 2 2, 1, 3 3 2, 3, 1 4 3, 1, 2 5 3, 2, 1 6 Thus in our example, the accomplice should display the cards in the following manner: firstly, the 5 of Spades, then the 2 of Diamonds and lastly, the 6 of Clubs ! b. Permutation Card Trick: Heres the trick: A magician asks an unsuspecting observer to randomly arrange 10 cards which are labelled 1 to 10 in a hidden face down manner. The participant does not show the arrangement of these cards to the magician, but does show them to the magicians accomplice. The accomplice looks at the ten cards and flips over six of the cards in a certain ordered manner to reveal their values to the magician. The magician immediately identifies the values of the four remaining unknown cards. How does the trick work? We first note that by applying the Pigeonhole Principle, we can show that in any permutation of 10 distinct numbers there exists an increasing subsequence of at least 4 numbers or a decreasing subsequence of at least 4 numbers. (refer next section of our discussion). These are the numbers that remain hidden in our trick. The magician will know that the sequence is increasing if the accomplice flips over the other six cards from the left to right and it is decreasing if the other six cards are flipped over from the right to the left. We will now proceed to explain the trick behind the game: The trick behind the game: Given any sequence of mn+1 real numbers, some subsequence of (m+1) numbers is increasing or some subsequence of (n+1) numbers is decreasing. We shall prove the result by Contradiction method. Assume that the result is false. For each number x in the sequence, we have the ordered pair (i,  j), where i is the length of the longest increasing subsequence beginning with x, and j is the length of the longest decreasing subsequence ending with x. Then, since the result is false, 1  £ i  £ m and 1  £ j  £ n. Thus we have mn+1 ordered pairs, of which at most mn are distinct. Hence by the Pigeonhole Principle, two members of the sequence, say a and b, are associated with the same ordered pair (s,  t). Without loss of generality, we may assume that a precedes b in the sequence. If a Thus, in our trick, we should have an increasing subsequence of at least (3+1) numbers or a decreasing subsequence of at least ( 3+ 1) numbers in a permutation of (33+ 1) distinct numbers! Here is an example of how the trick can be performed: Example Suppose the participant arranges the 10 cards in the following manner (value faced down from left to right): 3, 5, 8, 10, 1, 7, 4, 2, 6, 9. Upon careful inspection, the accomplice notices that an increasing subsequence can be 3, 5, 8, 10 while a decreasing subsequence can be 10, 7, 4, 2. If he decides to use the increasing subsequence, he should leave the first four cards untouched and flips the other six cards over in a leftward manner as shown: 1 7 4 2 6 9 Direction of flip The magician on realising that the four missing numbers are 3, 5, 8 and 10 and the leftward direction of flip, will thus proclaim the 4 hidden numbers to be 3, 5, 8, and 10 respectively! If the accomplice decides to use the decreasing subsequence, he should leave the cards bearing the numbers 10, 7, 4, 2 untouched and flips the other six cards over in a rightward manner as shown: 3 5 8 1 6 9 Direction of flip The magician on realising that the four missing numbers are 2, 4, 7 and 10 and the rightward direction of flip, will thus proclaim the 4 hidden numbers (from left to right) to be 10, 7, 4, 2 respectively! 8. Conclusion Although the Pigeonhole Principle seems simple and trivial, it is extremely useful in helping one to formulate and facilitate calculation and proving steps for numerous important Mathematical results and applications. We have included just a substantial amount of its applications in our project discussion. More importantly, we will like to show that a simple Mathematical concept like the Pigeonhole Principle does have numerous interesting and beneficial application in our daily life! ~ ~ ~ ~ ~ End of Report ~ ~ ~ ~ ~