Laura Tennant Senior Seminar Dr. Beatty Project Paper Due 4/27/12 The Math Behind 21 Ben Campbell, played by Jim Sturgess, is being interviewed by a scholarship committee member for a complete full-ride scholarship to Harvard Medical School, which Ben has already been accepted to, pending his graduation from MIT. Ben has a 4.0 GPA, got a 44 on his MCATs, is the president of a multiple clubs, and is working on a robotics project with his friends for a science competition. Despite all of Ben's amazing qualifications the interviewer tells Ben that 76 other applicants have just as amazing resumes. He tells Ben that he needs an outstanding life experience to make him "dazzle." Ben pleads with the interviewer that there is no way he can afford to go to Harvard Med without this scholarship and that's the only thing standing in his way. Ben and his friend are in a non-linear equations class where Professor O'Reilly, played by Kevin Spacey, is posing a few difficult questions to the class. The first question is regarding the Newton Raphson method of solving non-linear equations (which will be discussed in detail later). Ben answers it very well and Professor O'Reilly is impressed. He becomes interested in Ben. He then poses a question known as the Monty Hall Problem. (Will be discussed later, as well). One night shortly after, Ben is studying in an empty library. Another student in O Reilly s class comes to get him and tells Ben to follow him. They go to a class room where the rest of the crew is sitting in front of O'Reilly also known as "Mickey Rosa." They explain to Ben that they study counting cards and go to Vegas every weekend as a team to gamble and make money. They explain that what they do isn't illegal and that they want Ben to join because of his gift with numbers. He turns them down, saying he has a job and cannot join them. At Ben's workplace, a suit store, Jill, played by Kate Bosworth, comes by to ask Ben to join again, but he says no. In the next scene the secret card counting team is practicing, where they are all answering Mickey's questions wrong. Ben appears in the doorway and answers correctly. He joins the team. He learns the signs, symbols, method of counting, and all about what they do. They tell Ben he will have to go to a practice run. We see Ben enter an underground gambling ring and he looks nervous. He goes to a hot table as he is signaled and starts to play. He is winning money, when all of a sudden a bag is put over his head and he is dragged out of the room. Scared for his life, he freaks out when it is revealed it is the team, testing him. He passes and they tell him he will be leaving with them for Las Vegas the next day. Ben becomes the big player and wins lots of money going to Vegas each weekend. The team splits the weekend winnings and after a while, Ben has saved over $160,000. Security at the casinos begins to recognize Ben and the team, and see that they are losing money. One of the head security guys, Carl, gets mad and is determined to catch him. One night in Vegas, Ben decides that instead of counting cards, he is actually just going to gamble. He loses $200,000 and Mickey is furious. Mickey screams at the group and tells them they will pay and they need to find their own way back to Boston. After Ben convinces them, the team decides to stay and win the money back without Mickey. We learn here that Ben has now saved over $315,000 but his priorities have changed and he is becoming greedy. The team decides to go on with their plan to win back the money without Mickey and keep it for themselves. However, in the middle of the job, we see Mickey did not leave Vegas but is watching them inside the casino. Mickey gets on the phone and calls Carl and rats out the team. Carl comes to get Ben, and the rest of the team runs away leaving him. Carl and other security take Ben to an underground
room where they beat him up till his face is bloody. The next scene shows Ben leaving the casino the next day bruised, bloody, and with a black eye. Ben goes back to Boston, and finds out that Professor O'Reilly/Mickey is going to fail him in his course and he will not graduate, and therefore not be able to go to Harvard Med School. Ben's room was raided and destroyed and all of his money taken. He then makes a scene in his non-linear equations class by asking about a physicist who stole his student s ideas and ruined their lives. O'Reilly tells Ben and the class there was no proof so no one will ever know what happened - referring to their situation. Later, Ben seeks out Mickey and tells him they need to talk and that he will work with him again and win back all the money together. Mickey agrees and the team sets out to Vegas one last time, where this time both Ben and Mickey will be counting cards to win. They play for a while at the same table and are making lots of money, when security shows up. They throw all their chips in a bag as fast as they can before security begins chasing Ben, Mickey, and the team through the casino. They get to a point where they need to divide and Mickey tells Ben to give him the chips. Ben is hesitant but gives him the bag, and Mickey runs away. Outside you see Mickey get into a limo and drive off heading for the airport, with a smile on his face. He has scammed Ben again. He stole all his money he had saved, and now all the money Ben made that night that was going to replace it, or so he thought. A flashback shows that Ben knew Mickey was going to try and scam him, so Ben traded the bag of chips for a different bag right after they were approached by security when gambling together. Ben and Jill have the bag of chips and Mickey has a bag of chocolate gold coins. Another flashback to when Ben was getting beat up by Carl. This time you hear a dialogue between Carl and Ben, that basically says Carl has been chasing Mickey for years and if Ben can get Mickey to the casino so Carl can capture him, Ben can have one night of playing and keep all his winnings for himself without being bothered. Mickey soon realizes he was set up by Ben. He gets brought to the same basement room Ben was beat up in, and he is also strapped to the chair and beaten up. Jill and Ben think they have made it home free after setting up Mickey and getting his night of winnings when Carl appears behind them and tells Ben to give him the bag of chips. Ben expresses disbelief and horror because he needs the money and has been through so much to get it. Carl tells Ben he knows it s unfair but that his dreams will work out if he uses his brains and finds a way. He demands the chips and shows Ben that he has a gun. Ben gives him the chips, and he and Jill walk away. Ben tells bits and pieces of the story as a few feel good scenes occur and talks about how he admitted to his mom that he didn't get the scholarship. Ben stops talking and we see him sitting in the same scholarship interviewer s office as from the beginning. You realize that the movie was Ben telling him the entire story of winning money, getting it stolen, and being beat up, his friendships and experience. Ben says "Now how is that for a life experience? Dazzling enough?" The uptight interviewer has his jaw dropped to floor and the scene goes to Ben walking into a Harvard Med building. Now, here come the details about the math behind the movie: 1. Fibonacci Series: In "21", when Ben Campbell is celebrating his birthday, his cake reads 0, 1, 1, 2, 3, 5, 8, 13,... These are the first terms in the Fibonacci Series, which was used as an example in the book Liber Abaci published in 1202 by Leonardo Fibonacci. This is obtained by first writing the numbers "0, 1", then defining each subsequent number as the sum of the previous two numbers in the series. Thus, the third number in the series is 1 = 1+0, the fourth number is 2 = 1+1, the fifth number is 3 = 2 + 1, etc. The next number on the cake would be 21=13+8, for Ben's 21st birthday. Ben will have to wait until he is 34 = 21+13 for his next "Fibonacci birthday." One can define other Fibonacci Series by specifying different numbers in the first two slots. For example, the Fibonacci Series starting with "2, 5" is
2, 5, 7, 12, 19, 31, 50,... 2. The Monty Hall Problem: There are three doors, and behind one of them is a car, while behind the other two are goats. If you choose the door with the car behind it, you win the car. Now, say you choose Door 1. The host Monty Hall then opens Door 3, behind which is a goat. (He knows what is behind each door, and never opens the door with the car behind it.) Monty now gives you the choice: do you want to stick with Door 1, or switch to the other door? What should you do? Or does it matter? This question is posed to Ben Campbell by Professor O Reilly (Mickey Rosa) in the movie "21". Without hesitation Ben answers this correctly, which convinces Professor Rosa that Ben would be a good addition to their "card counting team". One solves this problem by comparing the probability of choosing the car if you stick with your original choice to the probability of choosing the car if you switch after Monty opens the one door. Note that the car has an equal probability of 1/3 of being behind Door 1, Door 2, or Door 3. First, suppose that your strategy is to stick with your original choice of Door 1. Then you only win if the car is behind Door 1, so that your probability of winning is 1/3. Next, suppose that your strategy is to switch doors. This now leaves three different cases: If the car is behind Door 1, Monty will open either Door 2 or Door 3 to reveal a goat. You switch to the other of Door 2 or Door 3, and in either case you switched to a door with a goat behind it (remember, the car is behind Door 1). If the car is behind Door 2, Monty will open Door 3. This is because he always opens a door with a goat behind it, and he can't open Door 1 because that was your original choice. So the only door you can switch to is Door 2, which is the door with the car behind it. Ding! You win! If the car is behind Door 3, Monty will open Door 2. This is because he always opens a door with a goat behind it, and he can't open Door 1 because that was your original choice. So the only door you can switch to is Door 3, which again is the door with the car behind it. Ding! You win! So if your strategy is to switch doors, you win 2/3 = 1/3 + 1/3 of the time. (Remember, the probability is 1/3 that the car is behind any particular door.) Therefore, a better strategy is to switch doors - the calculated probabilities indicate that you are twice as likely to win if you do this! Ben's correct answer in the movie "21" indicates that he is a good person for "counting cards". Not only does it show that he is clever, but it also demonstrates that he realizes that it is best to go with the choice which maximizes your probability of winning. This realization is essential to the success of "counting cards" for Blackjack. 3. The Newton-Raphson Method: In the movie, Professor O Reilly is lecturing on the Newton-Raphson method for finding the roots of f(x), when f(x) is a non-linear equation (That is the course he is teaching!). This was developed independently by Isaac Newton and Joseph Raphson in the 1600's. The idea is to make a guess for a root of the equation (call it x 0 ), then to use this guess to generate a value for x (call it x 1 ) which is even closer to the root than the original guess. This is done by drawing the tangent line to the function f(x) at x=x 0, and taking x 1 as the value for x at which this straight line goes through zero. (tangent line is determined by the derivative of f(x). By iterating this procedure over and over to generate x 2, x 3, etc, you obtain values which are better and better approximations to the
root. By calculating the approximate estimated error after every iteration, you are able to determine how many distinct digits are correct, as the estimated error continues to get closer to 0%. Unfortunately, the Newton-Raphson Method isn't always successful, although it is more likely to be if you make a good initial guess. The Movie 21 is filled with many more mathematical references (most do not need explanation as these do) that will catch the ear of any mathematically interested person that watches. That is the reason I enjoy the movie so much and continue to watch it any time it is on TV. Hopefully, this paper, along with my presentation, will motivate you to watch the movie again and you will catch all of the different math references throughout!
Sources: www.youtube.com http://www.npr.org/templates/story/story.php?storyid=89583891 www.imdb.com