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.

How to answer Walk me through your resume in a job interview

How to answer Walk me through your resume in a job interview You’re all excited for your interview and then you get the dreaded question. â€Å"Walk me through your resume,† the interviewer says. â€Å"Walk through it yourself!† you think. â€Å"Can’t you read?† We kid- of course, you can only think this. (Please, only think it!) It may seem like a ridiculous question, but think of it instead as an opportunity. You have a captive audience here to really fill in any holes or gaps on your resume and talk your interviewer through the bullet points while fleshing them out with the details they deserve! Take care, however, that you don’t let your own overconfidence trip you up. Be prepared to give a succinct, non-rambling answer. Also be prepared to have a smooth way of glossing over positions you put on for necessary padding in the hopes that the recruiter wouldn’t read those sections too carefully.Here are a few good tips for preparing to answer this common question.1. Be brief and articulate.Conci sion is your friend. Don’t just wing this answer- practice it with all your other interview prep answers. Remember, you’re probably also being evaluated here on your communication skills. Cut out the â€Å"um†s and set your resume in the context/tone you want without droning on too long about yourself. And whatever you do, don’t let yourself get bogged down with irrelevant extra details that have nothing to do with why your past will make you an ideal candidate for the job you want.2. Keep your eye on the prize.Your job in this answer is to put everything in your resume in the context of the job you’re applying for. Stick to the script! When you’re talking about your background, mine it for little gems that will show just how perfect your history makes you for this position. Explain your career so far in such a way that your interviewer will see how it perfect it makes you for the open job.3. Explain yourself.Some questions undoubtedly come up with any resume, no matter how bulletproof. Try to anticipate any questions or concerns your interviewer might have and have answers prepared. Explain your motivations and what went into the various decisions you’ve made along the way. Try to frame every negative or challenge into a positive. Show yourself to be scrappy and problem-solving when you go over any gaps and potential red flags, as well.4. Anticipate what they want to hear.As much as you want to practice a perfect answer, make sure to also think on the fly. Try to suss out what your interviewer is actually after and incorporate that into your answer. Tailor or fix what you’re about to say what you know they’re after.5. Stay classy.Paint yourself as the best person for the job, but don’t just shine yourself in best, brightest light and trash everyone else. Stay honest. Don’t throw anyone under any buses. Take responsibility for yourself and your actions, and don’t exaggerate you r successes while sweeping all weaknesses under the rug (even though you shouldn’t dwell too long on those)- you just won’t seem believable.6. Think big.When in doubt, frame your answer to be about how  this job is the perfect next step on your career path and how everything in the past has led up to this position.  Don’t forget to show the big picture and highlight your passion and enthusiasm for the position, in addition to your expertise.  Back everything you say up with actual bullet points on your resume. Stick to the script, show off without being too braggy, and the job should be yours!

The ultimate guide for matching your shirts and ties.

The ultimate guide for matching your shirts and ties. Thinking about spicing up your wardrobe from drab to fab? Maybe you should try switching up your shirts and ties. Replacing your shirts and ties with some fun colors and patterns might be just what you need to break you out of the boring office outfit. But before you go all pattern crazy here is what you need to know about mixing colors and patterns†¦and even patterns and patterns. With this simple guide, you will never have to second guess your outfit combinations.  Source [BeckettSimonon]

American political parties, relatively weak compared to similar Essay

American political parties, relatively weak compared to similar nations, why and what are consequences - Essay Example mparison to other highly industrialized society because in reality and practice, the American political parties have always had severe internal divisions. In a two-part system, ideal internal ideological unity could be nothing more than political picture. The political parties are plainly too immense and varied to encourage stable internal unity. Yet, integrated party control confers an electoral incentive for party leadership to downplay, or even vigorously steer clear of, those matters on which the party is divided to protect the outward show of cohesion. In the concern of electoral success and a significant party ticket, party member find motivating incentives to downplay intraparty conflict. Alternatively, it is in my own opinion that the capability of a divided government to provoke resentment rest partly in its capability to generate prospect for intraparty conflict to arise. Moreover, in a divided government system, the prevailing congressional party has the power to compel public concern of those matters most perceptive to the president’s party. Through placing conflict-ridden issues on the agenda, the party dominating Congress compels the president’s party to discuss matters its congressional members would vigorously steer clear of. The typical justification for the irresponsible nature of American political parties is that our very Constitution makes our party system what it is. Richard Hofstadter, the historian, put it in a few words: â€Å"We have a Constitution against parties† (Disch 2002: 62). However, it is also possible that federalism contributes to national and state parties that are primarily self-governing but it is also possible that federalism is the cause. The actual problem is in fact the failure of the political parties to attain an appropriate balance of power within the federal system, which demands a higher nationalization of party influence. Furthermore, it is unfortunate that there is no focus of party leadership in the American

Insurance Is Not Always Beneficial to Emplyees Essay

Insurance Is Not Always Beneficial to Emplyees - Essay Example In the United States, for example, many employees get Insurance policy through their employers where the employers help pay the Insurance policy. When the employer insures one, it is usually through a managed care plan that are health insurance plans that provide health care facilities to members at reduced costs by laying down contracts with medical facilities and health care providers. Employer based Insurance is often much cheaper as compared to buying individual Insurance, hence many people rather have their insurance bill cut from their salaries. The United States unlike other industrialized countries has a different case in that it does not have a national Insurance System. It is important to be insured because one cannot always tell when they are going to need the medical attention. There are vast advantages of getting employee-based health insurance. In the essay, I will look at health insurance in group verses individualized context, explore the advantages of employer based health insurance plan and discuss that insurance is not always beneficial. Group verses Individualized Insurance Employment based Insurance is an example of group based Insurance. ... e of group insurance policy is that it is much easier to obtain in that there are little procedures to be followed, which would in turn delay or even terminate the whole application process like inquiring about the health history of individual members. Group insurance policy is definitely the cheapest form of insurance available. However, as compared to group insurance, there are numerous advantages of individualized insurance plans. Firstly, one is able to customize the provisions they may want included or neglected in their package. They may also choose to regulate on the deductible amount of payment that the insurance company does. It also enables individual to choose the doctor that would be conducting their services. Here the price range may be elevated as compared to group insurers, but the services are better. It may also be exceptionally costly particularly when one runs a high risk being ill or has high health risk qualities. Advantages of Employment Based Health Insurance P lans A health plan would gain more if groups of individuals with low health risks were enrolled in it as compared to unhealthy list of individuals. In the United States where the employee health insurance benefit scheme is purely voluntary, there is a great chance that a group of highly unhealthy individuals are registered in the plan. Insurers would often prefer to enroll a group of individual as compared to the specific individual. Employment based insurance groups are one of the individual groups that are always considered to be formed for the sole purpose to offer their employees health insurance. It does not discriminate and whether the individual in the group posses a high health risk than the other, they will automatically qualify for the system. Not only are employers capable of

The Meaning of American Pie Essay Example for Free

The Meaning of American Pie Essay In the autumn of 1971 Don McLeans elegiac American Pie entered the collective consciousness, and over thirty years later remains one of the most discussed, dissected and debated songs that popular music has ever produced. A cultural event at the peak of its popularity in 1972, it reached the top of the Billboard 100 charts in a matter of weeks, selling more than 3 million copies. By identifying this great success it illustrates that it was no ordinary song. With its boldness, originality and it being thematically ambitious created uncertainty. Presenting the idea that we weren’t entirely sure what the song was about, provoking endless debates over its epic cast of characters. But however open to interpretation the lyrics may have been, the songs emotional resonance was unmistakable: McLean was clearly relating a defining moment in the American experience—something had been lost. Opening with the death of singer Buddy Holly and ending near the tragic concert at Altamont Motor Speedway, we are able to frame the span of years the song is covering—1959 to 1970—as the 10 years weve been on our own of the third verse. It is across this decade that the American cultural landscape changed radically, passing from the relative optimism and conformity of the 1950s and early 1960s to the rejection of these values by the various political and social movements of the mid and late 1960s. American Pie appears to chronicle the course of rock n roll, it is not, as is sometimes suggested, a mere catalogue of musical events. In using the cast of rock n roll players from the 1960s and setting them against the backdrop of Buddy Hollys death, they become polarized—metaphors for the clash of values occurring in America at this time: Holly as the symbol of the happier innocence of the fifties, the rest as symbolic of the sixties growing unrest and fragmentation. And as each verse sums up chronological periods in time—the late 1950s, 1963-66, 1966-68, 1969, 1970—another blow against the happier innocence of another era is registered: another day the music dies. Verse 1 of American Pie looks back from the early seventies and introduces the catalyst for the story about to unfold. â€Å"A long long time ago I can still remember how the music used to make me smile†. The narrator here is nostalgic for a simpler and more optimistic kind of music—a music that can make people smile, and that could help them forget their troubles—and a music that very much represents the happier optimism of the 1950s in America. â€Å"But February made me shiver†, he also identifies Buddy Holly by the month in which he dies. Hollys passing had a profound effect on him, which is displayed throughout the song. The day the music died† this reflects and supports the idea that the day the music died becomes the day the innocence and optimism within America died The chorus is the primary key in understanding American Pie as the theme of America’s lost innocence is clearly stated. â€Å"So bye bye Miss American pie† Miss Amer ican Pie* is as American as apple pie, so the saying goes; she could also be a synthesis of this symbol and the beauty queen Miss America. Either way, her name evokes a simpler time in American life when these icons held more meaning. She is the America of a passing era, and he is bidding her farewell. Drove my Chevy to the levee alludes to a drive along a levee mentioned in a series of popular 1950s Chevrolet television commercials sung by Dinah Shore and which serves as a signpost to that era—just as the Chevrolet itself is a familiar icon of 1950s America. Also, given that a drive to a levee carries the suggestion of romance in a car, we can almost see him on a date here. But the date is over, the levee is dry—someone he once loved has betrayed him; something that once gave him sustenance has evaporated. This’ll be the day that I die† is a rewording of the line â€Å"cause that’ll be the day when I die† from the chorus of Holly’s hit that’ll be the day. This signifies McLean’s way of both mourning the death of that music and way of life, and pointing to Holly as his symbol of it. Verse 2 erupts with the idea that the narrator reaches little further back in time to the days of his youth, the late 1950s—a time of sock hops, pickup trucks and pink carnations—as he courts a woman who ultimately spurns him. This is a fickle lady here, and the narrator questions her loyalties. And can you teach me how to dance real slow? † This is a romantic dance. He is courting her. The slow dance itself is yet another reference to the fifties and the kind of dancing that went out of fashion in the following decade; it also alludes to the slower pace of life in America at this time. This verse helps us to further identity Miss American Pie, whose brief introduction in the chorus needed this additional exposition; and which, along with verse 1 and the chorus, also serves to establish the 1950s as the reference point for the res t of the song. In so doing, McLean characterizes the period primarily through its musical symbol (Holly), using him and the music (those rhythm and blues) as a metaphor for the innocence of the times, and a sacred thing. The day the music died now takes on the significance of a lost faith in the values of a passing era and the sorrow the narrator feels at their passing: blow number two. Having personified America as a woman, Bye bye Miss American Pie now more clearly becomes a farewell to the America he once knew.

Pompey :: essays research papers

Pompey the Great Gnaeus Pompeius, better known as Pompey, was born on September 29, 106 BC. He was four years older than Julius Caesar. Pompey’s father was a rich Roman noble, who was elected to the consul in 89 BC. Pompey distinguished himself as a great leader early in his life. In the civil war between Gaius Marius and Lucius Sulla, Pompey sided with Sulla. Sulla, with the help of Pompey, made some vary impressive defeats in Africa and Sicily. In 79 BC Sulla resigned and died the next year. Two of his patrons, who had fought for him, Pompey and Marcus Crassus, moved to leading military positions in the seventies.   Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Crassus and Pompey fought together in a battle against a Marian rebel, Quintus Sertorius, and a slave rebellion lead by Spartacus in Italy. They returned, having won, in 71 BC. Pompey then spent time campaigning successfully in Rome before he was elected to consul, with Marcus Crassus for the year 70 BC.  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  After Pompey served his time on Consul he was given command over the Mediterranean, where he did what nobody else had successfully done before. He rid it of Pirates. Pompey, then, went to various places, establishing an ally of the King of Armenia, capturing Jerusalem, and making Syria a Roman duty. Pompey was a great general, but not a very good politician.  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  In 59 BC Pompey returned to Rome to find that tensions with himself and Crassus had grown. Both Crassus and Pompey had large armies, but also pieces of the city that were loyal to them. Cicero, the leader of the sena te, allied himself with Pompey through great flattery. Cicero told Pompey that he must be the protector of the republic. Crassus had other plans, and by 57 BC both men were in Italy with their armies. Before war broke out Julius Caesar stepped in.  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Caesar being a neutral negotiator used these well-known talents and convinced Pompey, Crassus, and Cicero to meet. The men worked out an agreement. This settlement had never been made before among the leaders of Rome. Caesar convinced Crassus and Pompey to join their power and influence with his own. Caesar was a successful leader of Gaul at this time. So the three agreed, and formed what is today known as the First Triumvirate. During this time Pompey married, most likely for political reasons, Julia, Caesar's daughter.  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚     Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Two of the three men returned to Rome and forced the Senate to obey them.