Number Stairs Essay Example

Number Stairs Essay Part 1: To investigate the relationship between the stair total and the position of the stair shape on the grid, for other 3-step stairs. The number in the bottom left corner of the stair shape labels the position of the stair shape. The aim of this part of the investigation is to find a formula to work out the s-total (stair total) by knowing the s-number (stair number).45353625262725 + 26 + 27 + 35 + 36 + 45 = 194s-number = 25s-total = 1942111121231 + 2 + 3 + 11 + 12 + 21 = 50s-number = 1s-total = 502212132342 + 3 + 4 + 12 + 13 + 22 = 56s-number = 2s-total = 562313143453 + 4 + 5 + 13 + 14 + 23 = 62s-number = 3s-total = 622414154564 + 5 + 6 + 14 + 15 + 24 = 68s-number = 4s-total = 68Another observation here is that in each column of the stair shape, the difference between these rows of numbers in the same column, is 10. (Apart from the last column, which only consists of 1 number.) The difference between these numbers is 10 because in the grid, each row contains 10 numbers.Table of Results:s-numbers-total150+ 6256+ 632+ 646825194To go from one term to the next, simply add 6 each time. However, this does not show the relationship between the s-number and the s-total. This sequence (adding/subtracting a number each time), is known as an arithmetic sequence. A formula to find out the relationship between the s-number and the s-total is: first term + common difference x ( n 1 ). The first term here is 50, and the common difference is 6.50 + 6(n 1)= 50 + 6n 6= 6n + 442515165675 + 6 + 7 + 15 + 16 + 25 = 74(6 x 5) + 44 = 74s-number = 5s-total = 742616176786 + 7 + 8 + 16 + 17 + 26 = 80(6 x 6) + 44 = 80s-number = 6s-total = 8046363726272826 + 27 + 28 + 36 + 37 + 46 = 200(6 x 26) + 44 = 200s-number = 26s-total = 200After testing the formula to see if it works, a proof must be presented to show that it works for any s-number (with a few exceptions see Conclusion). If the s-number is represented as n:nAs mentioned in the aim, the bottom left corner square is the s-num ber (or in this example, known as nnn + 1n + 2Because the numbers in the grid increases by 1 each time, I can say that the two numbers, in the same row, increases by 1, thus the diagram; n + 1 and n + 2.n+20n+10nn + 1n + 2As mentioned previously, the numbers in the same column have a difference of 10. Thus, n + 10 and n + 20 as shown in the diagram. I can then show: (v)n+20n+10nn + 1n + 2If I add all these values (inside the squares) together:n + (n + 1) + (n + 2) + (n + 10) + (n + 11) + (n + 20)= 6n + 44*Proof 1: Note that there are 6 ns in this equation, because there are 6 squares.Thus, I have proven that my formula 6n + 44 works for every stair shape in that grid size. I can carry this investigation further by investigating the relationship between the stair totals on other number grids.Part 2: To further investigate the relationship between the stair totals and other step stairs on other number grids. I have decided to investigate the relationship between the stair totals on ot her number grids when the s-number stays constant because it will be easier to observe the relationship between the s-grid (stair number grid) and the s-total.7451231 + 2 + 3 + 4 + 5 + 7 = 22s-grid = 3s-total = 229561231 + 2 + 3 + 5 + 6 + 9 = 26s-grid = 4s-total = 2611671231 + 2 + 3 + 6 + 7 + 11 = 30s-grid = 5s-total = 3013781231 + 2 + 3 + 7 + 8 + 13 = 34s-grid = 6s-total = 34Table of Results:s-grids-total322+ 4436+ 4530+46341050In the table, it shows that the s-total increases by 4 each time as the s-grid increases by 1. (When g = grid size,) this is because:n+2gn + gn+g+1nn + 1n + 2; Previously, I noticed that the difference between each row in the same column is the grid size.n + (n + 1) + (n + 2) + (n + g) + (n + g + 1) + (n + 2g)= 6n + 4g + (4)**Note: 6n + 4g because there are 4gs in the formula (as shown above).Now that I have found out the relationship between the s-number, s-grid, and the s-total, I can carry this investigation further by investigating the relationship betw een the other step stairs, s-number, s-grid, and s-totals.I will investigate the relationship between the s-number, s-total, s-grid and s-shape (stair shape). The s-shape will be investigated by enlargement; 1 square increase in length, and 1 square increase in width.nThe s-number remains at the bottom left corner of the stair shape. Instead of 6 squares, there are 10. Based on my previous observations, I can predict the relationship between this s-shape, the s-number, s-grid, and the s-total as: 10n + 10g + 10.I can check my prediction:n+3gn+2gn+2g+1n+gn+g+1n+g+2nn+1n+2n+3n+(n+1)+(n+2)+(n+3)+(n+g)+(n+g+1)+(n+g+2)+(n+2g)+(n+2g+1)+(n+3g)= 10n + 10g + 10I shall investigate the next stair shape, which will have a height of 5 squares, and a width of 5 squares, a total of 15 squares.n+4gn+3gn+3g+1n+2gn+2g+1n+2g+2n+gn+g+1n+g+2n+g+3nn+1n+2n+3n+4n+(n+1)+(n+2)+(n+3)+(n+4)+(n+g+)+(n+g+1)+(n+g+2)+(n+g+3)+(n+2g)+(n+2g+a)+(n+2g+2)+(n+3g)+(n+3g+1)+(n+4g)= 15n + 20g + 20Observation: I have noticed that the number of squares in each stair shape has a link with triangle numbers. For example: a triangle number sequence is as follows:When the stair shape size was 3, it has 6 squares. When the stair shape size was 4, it has 10 squares, and when the stair shape size was 5, it has 15 squares. Just like the diagram on the previous page, the pattern for the stair shape size is the same pattern as triangle numbers. This is because the stair shape is a triangle:s-sizeno. of squares23+ 336+ 4410+ 5515+ 6621I can predict the number of squares they are for any s-size, following the triangle number sequence.The formula for the triangle number sequence is:(Where s is the s-size.)I should test this formula to check if it works. I shall try it on an s-size of 7.7+ 6+ 5+ 4+ 3+ 2+ 1Number of squares in this diagram: 28Since I have successfully tested out that the relationship between the s-size and the number of squares in that s-size is the same as triangle number sequence; thus the formula to work out the s-total when the s-number, s-grid, and s-size is known, can be currently worked out as:s-total = (Note: X is the unknown)In this case, I have used the Xs twice, only because they are of equal value. By putting the s-size value and the formula in a table, I could see why I used the Xs twice in the formula.s-sizeformula36n + 4g + 4410n + 10g + 10515n + 20g + 20The co-efficient has the same value as the constant.I then needed to find to how to find out the co-efficient and the constant. Because there doesnt seem to have a constant difference between the co-efficient, I shall use the method changing differences to find the formula for the co-efficient. There is a formula to find the changing difference known as the Gregory Newton Formula.Firstly, I would need to find the value of the co-efficient when the s-size is 0. Obviously, if the s-size is 0, the s-total will be 0, as there are no squares to add up its value. To show how I worked out the difference in order to use th e Gregory Newton Formula, I made a table of differences:s-sizeX1st Difference2nd Difference3rd Difference4th Difference00+ 010+ 1+ 1+ 121+ 2+ 0+ 3+ 134+ 3+ 0+ 6+ 1410+ 4+ 10520After producing the table of differences, I was able to carry out the Gregory Newton Formula and found out how to find X:However, I must test this formula to see if it works. I shall test it out on an s-size of 6.n+5gn+4gn+4g+1n+3gn+3g+1n+3g+2n+2gn+2g+1n+2g+2n+2g+3n+gn+g+1n+g+2n+g+3n+g+4nn+1n+2n+3n+4n+5n+(n+1)+(n+2)+(n+3)+(n+4)+(n+5)+(n+g)+(n+g+1)+(n+g+2)+(n+g+3)+(n+g+4)+(n+2g)+(n+2g+1)+(n+2g+2)+(n+2g+3)+(n+3g)+(n+3g+1)+(n+3g+2)+(n+4g)+(n+4g+1)+(n+5g)= 21n + 35g + 35It appears that this section of the formula works. Now the new current formula is:s-total =Important: However, this formula has its limitations. For example, the step stair may be placed on a certain area that causes the step stair to be off the grid. This means that the formula cannot calculate the s-total as there are no numbers off the grid. Per haps another limitation can be found if it cannot calculate negative numbers.I shall test it out on an s-size of 7, s-grid of 8, and the s-number is 20.6860615253544445464736373839402829303132332021222324252620 + 21 + 22 + 23 + 24 + 25 + 26 + 28 + 29 + 30 + 31 + 32 + 33 + 36 + 37 + 38 + 39 + 40 + 44 + 45 + 46 + 47 + 52 + 53 + 54 + 60 + 61 + 68= 1064Now to test the formula out on negative numbers:1314151617187891011121234560-1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17(-12)+(-13)+(-14)+(-6)+(-7)+(0)= -52It appears that negative numbers do not work with this formula, thus the formula is limited to positive integers and that the stair shape must not have any part of it exceeding the grid. However, the formula works excluding the limitations. In conclusion, the relationship between the s-number, s-grid, s-size, and s-total is:s-total =(With certain restrictions as mentioned above)However, this formula can be simplified because as previously mentioned, the Xs were in 2 parts of the formula, thus it can be simplified. The first part of the equation where the formula for triangle number lies, can also be simplified to:An extention of this investigation, could be to investigate the relationship between negative numbers, or transformation of the stair shape.

Abortion is a right choice. essays

Abortion is a right choice. essays We live in a nation built on the idea of freedom, freedom of choice and freedom of expression, yet we are not free. After all, different people have different attitudes toward abortion. Like wise, the oppositions of society create constricts for women seeking abortions. They maintain that since the unborn child in the womb is a human life, abortion is a kind of guilt, which deprives of a human right to life. Certainly, they go on to point out that abortion should be banned by the law in order to protect these human lives. Of course, the main questions that everyone must face are which areas of behavior to leave to private choice and which to regulate in the interest of virtuous. However, abortion is a very serious and complex issue. This issue will not just end with laws. Therefore, the fact of the conflict is not being concern of its legal status abortion is still going to exist. Notably, still have women who are going to seek them with the help of men to arrange them. Moreover, wit h the doctors and some not qualified who will still perform them. It cannot be denied, this is reality because before 1973 a lot of women died from illegal abortions. Currently, abortion is legal, however, if histories were to repeat itself, and abortions are considered illegal once again, the results would be appalling. Not only would women turn to insalubrious secretive abortions and self induced abortions, the psychological pain and scars would be considerably more awful than those of a legal abortion. In a word, every woman is permitted to the right of privacy in dealing with her own body if women are to be denied this right, history will repeat itself jeopardizing the lives of millions of women. Formerly, legalizing abortion will guide women to have safe conditions, with qualified doctors and proper care. By the same reason, if abortions were no longer an option, womens health issues would arise. Even though, the government will still be able to ...

Using the PHP Rand() Functions to Generate Random Numbers

Using the PHP Rand() Functions to Generate Random Numbers The rand() function is used in PHP to generate a random integer. The rand() PHP function can also be used to generate a random number within a specific range, such as a number between 10 and 30. If no max limit is specified when using the rand() PHP function, the largest integer that can be returned is determined by the getrandmax() function, which varies by operating system.   For example, in Windows, the largest number that can be generated is 32768. However, you can set a specific range to include higher numbers. Rand() Syntax and Examples The correct syntax for using the rand PHP function is as follows: rand(); or rand(min,max); Using the syntax as described above, we can make three examples for the rand() function in PHP: ?phpecho (rand(10, 30) . br);echo (rand(1, 1000000) . br);echo (rand());? As you can see in these examples, the first rand function generates a random number between 10 and 30, the second between 1 and 1 million, and then third without any maximum or minimum number defined. These are some possible results: 20442549830380191 Security Concerns Using Rand() Function The random numbers generated by this function are not cryptographically secure values, and they should not be used for cryptographic  reasons. If you need secure values, use other random functions such as random_int(), openssl_random_pseudo_bytes(), or random_bytes() Note: Beginning with PHP 7.1.0, the rand() PHP function is an alias of mt_rand(). The mt_rand() function is said to be four times faster and it produces a better random value. However, the numbers it generates are not cryptographically secure. The PHP manual recommends using the  random_bytes() function for cryptographically secure integers.

Pride and Prejudice Essay Example | Topics and Well Written Essays - 1000 words

Pride and Prejudice - Essay Example Elizabeth had asserted her love, and her experience can also be useful for women today. Austen's use of irony in her novel offers important insights to her attitude about life through the experiences of her characters. This paper will discuss what Jane Austen had to say about women, class mobility, and marriage based in the experiences of Elizabeth Bennet in Pride and Prejudice. In addition to that, the nature of Elizabeth Bennet's experience and lessons she "learned" from it will be estimated. The experience of women today and that of Elizabeth Bennet will be analyzed, as well as the reasons of Elizabeth Bennet's experience relevance for women position in our days. 2. Elizabeth Bennet is one of the main protagonists in Austen's Pride and Prejudice. Her experience in the novel is largely influenced by her nature and her personality, as well as appropriate traits of her character which allowed her to win her real love and destiny. Elizabeth is the second daughter in the family of the Bennets and the most clever and intelligent among them. Her positive traits allowed her to resist current bourgeois norms and traditions and to be higher than these norms. Mr Darcy noticed her charm and evaluated her as a woman who had interesting and unique individuality, but at their first meet they didn't had any romantic feelings towards each other - so, first impression can be not true. Darcy said that "She is tolerable, but not so handsome enough to tempt me; I am in no humour at present to give consequence to young ladies who are slighted by other men" (Chapter 3, p.7). It happened because of Elizabeth Bennet's pride and Darcy's prejudice about her social bac kground. So, in that moment Jane Austen depicted about the common social norms which were spread between men and women in that time, and those norms often asserted women to marry men who would provide high social status for their future wives. There are some characters in the novel who made different obstacles towards the relationships development between Elizabeth and Darcy, such as Mrs Bennet, Lady Catherine, Miss Bingley and Wickham, and the motifs which these people followed interfered to romantic relationships between Elizabeth and Darcy. Lady Catherine tried to control Darcy who was her nephew, Mrs Bennet tried to get married her daughters, including Elizabeth, by all means, and it led to numerous foolishness displayed by Mrs Bennet. Wickham deceived Elizabeth telling her that Darcy had taken his money, and that caused her mistrust to Darcy: "How abominable! I wonder that the very pride of this Mr. Darcy has not made him just to you! If from no better motive, that he should no t have been too proud to be dishonest - for dishonesty I must call it" (Chapter 16, p.60). So, Austen has showed that there are many obstacles for people who love each other to be together, and that there are some women whose motifs for marriage are not love but social class advantages. The author tried to show that love and true affections of those who really love each other are able to be higher than any obstacles. But the author also showed that social class and aspiration towards money and success can be a basis for marriage for

The importance of statistics and consumer research in marketing Essay

The importance of statistics and consumer research in marketing - Essay Example In this way, both consumer research and statistics are found involved in marketing operation. This study is going to explore of how consumer research and statistics are important to marketing. The importance in terms of application and the outcome will be explored and described in this study. The objective here is to understand the relationships between marketing and consumer research and between marketing and statistics altogether. Importance of Consumer Research and Statistics in Marketing During marketing operation, strategists and marketers are keen to understand their consumers. They are deliberately in search of knowing their consumers’ perception, behaviors and their buying attitudes which help the marketers to bring effective marketing campaigns and strategies (Roosi and Allenby). For this comprehensive knowing and understanding, marketers plan the operation of consumer research which digs out all the information pertinent to consumers and their behaviors. Decisively, consumer needs and wants are uncovered through market consumer research (Mazzocchi). Consumer research basically gives the manifestation about how customers behave and what are their levels of motivations towards specific products or services. This research sets the direction for marketers that they bring a comprehensive marketing strategy, which is closer to consumers’ behaviors and their buying choices (Mazzocchi 10). ... Marketers conduct consumer research in order to understand the dynamics of a particular market. Definitely, each market has a distinctive background and which modifies or changes with the aspect of time and with the changes in attributes and characteristics of consumers present in that market (Mazzocchi). For understanding the varying dimensions of a market, the varying dimensions of consumers are important to be known, which is only possible by conducting effective consumer research and consumer diagnosis. Without studying the market and consumers trends, a marketer cannot devise a sound workable marketing strategy (Roosi and Allenby). The strategy, which can impact the dynamics of a market and can attract diverse market consumer, may certainly require a comprehensive consumer research. For all these reasons, marketers consider consumer research as a significant part of marketing planning. They realize that without consumer research, marketing plans cannot be effectuated or even ini tialized. A well-organized consumer research is actually a source to ideate a well-incorporated marketing plan. This indicates that strategically and distinctively there is a significant relationship between consumer research and marketing. Without consumer research marketing is incomplete and similarly without marketing consumer research is reasonless or futile. This shows a certain sort of connection between marketing and consumer research. Both are part of each other and so are dependent on each other (Mazzocchi 30). In marketing, the core objective of consumer research is to gather consumer related information. This process of information collection in consumer research is made possible by means of statistical models and techniques. There are different

Ethics of Biobanks

Ethics of Biobanks Biobank is a large collection of biological information and tissue samples kept for research purposes. It is also a powerful tool used in the study of diseases. It is an important resource in supporting different types of contemporary research such as personalised drug and genomics. Biobank enable scientist to have cross purpose research studies in which data derived from samples in biobanks can be used for multiple researches. E.g. Biobanks can enable scientist identify disease biomarkers by using large collections of samples which represent hundreds of thousands of people. Its been shown that before biobanks was invented little or known was known about different disease and biomarkers and scientist struggled to find enough samples to know what sort of disease they are dealing with. Although its not all good news for the use of Biobanks due to research ethics and medical ethics. This issues were raised because of PRIVACY whereby operating biobanks without the knowledge of governing bodies and policies could be bad for the societies that take part in Biobank programs There are types of biobanks, Tissue banks and Virtual biobanks and population banks, before I explain the types Im going to explain a lot more about biobanks Biobanks incorporate cryogenic storage facilities for samples in which it can be an individual refrigerator or a big warehouse refrigerator. They are kept up to standard by the hospital, pharmaceutical companies and universities etc. Disease oriented biobanks may be classed by design and purpose because this biobanks collect information or samples representing different forms of diseases in which it can be used to also find a biomarker associated with a specific disease. Population based biobanks are big biobanks that collect large samples from large numbers of people in a community. This is done to look for biomarkers for disease in a general population. Tissue Banks- Store and harvest human tissue for transplantation, stem cell and researches based on tissue and cells Virtual biobanks samples are collected and termed to meet national regulations and integrate epidemiological cohorts Population banks they store organic material associated with clinical, lifestyle and environmental data. Biobank ethics There are many roles which comes into effect when researchers wants to collect a human specimen for research and storing it. The issues that comes into effect are the right of the participants to be private, ownership of the specimen and where the data is derived from. Also how far the donor can consent to the research study should be considered and to which extent the donor can far in sharing research results. The main issue is that biobanks collect sample and data for different future research and it is not easy to get a specific consent for any single research. Biobank controversies[1] issue consensus controversy notes Commercialization Different aspects of biobanks serve public, private, commercial, and non-commercial interests. How can policymakers set guidelines to fairly balance public, private, commercial, and non-commercial interests? Who owns biological specimens and data derived therefrom? When biobanks and related projects are publicly funded, the result will benefit private industry. To what extent is this outcome satisfactory? (Social Fairness). It may also undermine public trust in biobanks projects. It may skew research agenda in favour of research projects which are more profitable and compromise necessary but not profitable research. discrimination, including Genetic discrimination Biobanks should prevent donor communities from facing discrimination as a result of participating in a Biobank project Research reveals private information and release of it may cause participants to face discrimination. What responsibility does the Biobank have to mitigate the problem? Participants may reveal their own information because of participation in a Biobank and subsequently face discrimination. What responsibility does the Biobank have to mitigate the problem? informed consent Donors to biobanks need a consent process adjusted specifically to biobanks. What breadth of consent should biobanks have? [2] Institutional review board It would be nice to have a robust governance system before biobanks are created. How will a good governance system be designed? The oversight institution reviewing biobanks should be independent of the Biobank. Where should checks and balances be? An individual organization needs multinational support to do international research. Who should govern when research spans different countries with different legal and personal rights standards? Privacy for research participants Donors should have their specimens sufficiently anonymised. A specimen by nature includes some data about donors how much anonym zing is sufficient? [3][4] Donors have some right to return of results. How does one return results to anonymised donors? [3][5] Donors have a right to withdraw from research. Specimens can be destroyed, but to what extent should anonymised data which has already been shared be withdrawn? [5] Data derived from specimens should be shared. Who gets access and how much? [3] Changing technology makes it difficult for researchers to say how safe participant information is. What protections can be promised? [6] Return of results Donors have a right to know the purpose of a Biobank and what results it generates When should all donors share general information and when does each donor have a personal right to personal information? Public consultation Everyone wants the researchers and community to work together. What resources should be spent doing outreach, and how much involvement does the community want, and what role should the community have? Communities should participate in writing laws, standards, and policies for research. How can communities be encouraged to participate, who represents the community, and how much involvement should there be? Patients should be involved when there is research on diseases. When people are desperate because of a disease, to what extent can they participate fairly without feeling obligation to support research? Communities which donate specimens to a Biobank should have special involvement in their Biobank. What kind of involvement? Resource sharing Research efficiency increases greatly when resources are shared. How should beneficiaries share costs? This is especially problematic when a Biobank is a national resource and another country wants access to it. Results of studies should go to the widest possible audience. When should this happen and in what way? Can results be released with commercial licensing for use?

A Million Little Pieces Character List

A Million Little Pieces Characters James (1:1) Main character, miserable, self centered, drug addict Mom (1:2) James mother, she’s always upset, sad, she cries a lot Dad (1: 2) Is in charge of most family affairs, happy, concerned Nurse (1:8) she wears all white, smiles a lot administers the shots Men in White (1:11) the men in white take him away Doctor Baker (1:15) kind eyes, rehab doctor, helpful James’s love interest Lilly (1:18) black hair and blue eyes, drug addictLilly’s Dad (1:23) Left Lilly’s family when she was four Lilly’s Mom (1:23) Heroin addict, prostitute Lilly’s Grandmother (1:23) pays for Lilly’s, cared for Lilly as a child, dies while Lilly is still in rehab Roy (1:23) James roommate, follows the rules Larry (1:25)35 southern accent, short, alcoholic Warren (1:25)50, tall, thin, well dressed John (1:26) nervous and hypersexual ninja, addicted to cokeKen (1:28) Unit recovery counselor, nice at first but ends up being a wful Hank (1:35) the driver who works at the rehab, James friend, messy looking old man with white hair and blue eyes Dentist Stevens (1:36) does James surgery, he doesn’t use pain killers Amy (1:43) James sober friend Lucinda (1:43) James sober friend Courtney (1:43) James sober friend Lincoln (1:53) Unit supervisor, he hates James Joanne (1:53) Staff psychologist, secretly dating hank Ed (1:56) short man, nosey, blue collar worker Ted (1:76) tall man, deep southern accent Bill (1:77) the founder of AAMichelle (1:80) one of the only people who comes to visit James in rehab, James’ sober friend Bald Man (1:85) Alcoholic, has a wife and two kids, cries during a group meeting and gets made fun of Mickey (2:121) the gangster that Leo looked up to as a child, adopted him when he was a teenager, married to Geena Geena (2:121) Mickey’s wife and Leonard’s adopted mother, very sweet Eric (2:125) Roy’s friend who tells the counselors that Roy picked the fig ht Julie (2:126) James friend who comes to see him in rehab, very for giving Kirk (2:126) James friend, who comes to visit him in rehab, Matt (2:161) Featherweight Champion, addicted to crackDaniel (3:247) Counselor at family rehab center Sophie (3:259) Addict, alcoholic, married to Tony, is in the same rehab as James Michael (4:353) James friend, one of the guys who go with James to the bar for his first time when he’s out of rehab Kevin (4:424) James friend, one of the guys who goes with James to the bar for his first time when he’s out of rehab Bob (4:424)James friend, one of the guys who goes with James to the bar for his first time when he’s out of rehab Chapter One- (112 pages) James is a drug addict who is angry, sad and on the verge of death.In order to stay alive he must learn to live a sober life so his family sends him to rehab. In the rehab facility he finds a group of patients he fits in with and gets in a fight with Roy. Chapter Two- (113pages) Jam es continues rehab and is doing well. He starts to make friends with Leonard and they help each other through rehab. James also begins to meet secretly with a girl named Lilly even though it’s against regulations. James has an older brother that starts coming to see him on visiting days.Chapter Three- (113 pages) James begins family counseling and is having a lot of trouble dealing with his parents. Meanwhile, Lilies grandmother gets sick so Lilly leaves the rehab facility and relapses. Despite the consequences James goes after her, he does not relapse and they return together. Chapter Four- (69 pages) Aside from having nightmares about using drugs James is doing well in rehab and Lilly is recovering. James is released from rehab, shortly after Lilly’s grandmother dies and Lilly commits suicide. James is doing well and has yet to relapse.