State those questions here. Maybe you will inspire someone else to tackle those questions. Maybe you will work on them yourself at another time. By posing them here, you participate in advancing the field of mathematics. You place your work within a larger framework of inquiry. As with any research paper, you must give credit to the people whose work you used in writing your report.
Include articles and books that you used. It is also appropriate to cite articles and other material that you found on the World Wide Web. There is more than one accepted form for referencing materials; here we provide examples using the APA American Psychological Association style. For more complete instructions you may find one of the APA Style guides helpful.
Most libraries have a copy. Disquisitiones arithmeticae English edition. The art of problem posing. Mathematical thinking and problem solving. The role of definition. Mathematics Teaching in the Middle School 5, 8. Pinpointing a Polar Bear. Scientific American, , p. Aoki, Naomi , December Genetic mutation gives hope to those with osteoporosis.
Boston Globe , pp. Journal of Analytical Psychology , 38 1 , Abstract from SilverPlatter File: Sample Bibliographic Entries 4th ed. Retrieved February 12, , from the World Wide Web: In the appendices you should include any data or material that supported your research but that was too long to include in the body of your paper.
Materials in an appendix should be referenced at some point in the body of the report. The answers to combinatorial questions concerning unfriendly customers at a morning coffee shop are found. It is also proven that if there are n seats and c customers who refuse to sit next to each other at a circular table, then there are ways for them to sit. In a certain coffee shop, the customers are grouchy in the early morning and none of them wishes to sit next to another at the counter.
Suppose there are ten seats at the counter. How many different ways can three early morning customers sit at the counter so that no one sits next to anyone else? What if, instead of a counter, there was a round table and people refused to sit next to each other?
I am assuming that the order in which the people sit matters. So, if three people occupy the first, third and fifth seats, there are actually 6 3! I will explain more thoroughly in the body of my report. At first there are 10 seats available for the 3 people to sit in. But once the first person sits down, that limits where the second person can sit. What confused me at first was that if the first person sat at one of the ends, then there were 8 seats left for the second person to chose from.
But if the 1 st person sat somewhere else, there were only 7 remaining seats available for the second person. I decided to look for patterns. By starting with a smaller number of seats, I was able to count the possibilities more easily. I was hoping to find a pattern so I could predict how many ways the 10 people could sit without actually trying to count them all.
I realized that the smallest number of seats I could have would be 5. So, I started with 5 seats. I called the customers A, B, and C.
As I said in my assumptions section, I thought that the order in which the people sit is important. Maybe one person prefers to sit near the coffee maker or by the door. These would be different, so I decided to take into account the different possible ways these 3 people could occupy the 3 seats shown above. I know that ABC can be arranged in 3! So there are 6 ways to arrange 3 people in 5 seats with spaces between them. But, there is only one configuration of seats that can be used.
Next, I tried 6 seats. I used a systematic approach to show that there are 4 possible arrangements of seats. This is how my systematic approach works:.
Assign person A to the 1 st seat. Now, person C can sit in either the 5 th or 6 th positions. Next suppose that person B sits in the 4 th seat the next possible one to the right. That leaves only the 6 th seat free for person C.
These are all the possible ways for the people to sit if the 1 st seat is used. Now put person A in the 2 nd seat and person B in the 4 th. There are no other ways to seat the three people if person A sits in the 2 nd seat. So, now we try putting person A in the 3 rd seat.
If we do that, there are only 4 seats that can be used, but we know that we need at least 5, so there are no more possibilities. I continued doing this, counting how many different groups of seats could be occupied by the three people using the systematic method I explained.
Then I multiplied that number by 6 to account for the possible permutations of people in those seats. I created the following table of what I found. Next I tried to come up with a formula. I decided to look for a formula using combinations or permutations. Since we are looking at 3 people, I decided to start by seeing what numbers I would get if I used n C 3 and n P 3.
Surprisingly enough, these numbers matched the numbers I found in my table. However, the n in n P r and n C r seemed to be two less than the total of seats I was investigating. Given n seats at a lunch counter, there are n -2 C 3 ways to select the three seats in which the customers will sit such that no customer sits next to another one. There are n -2 P 3 ways to seat the 3 customers in such a way than none sits next to another.
After I found a pattern, I tried to figure out why n -2 C 3 works. First we pick three of them in which to put people, without regard to whether or not they sit next to each other. It would look like this:. This procedure guarantees that two people will not end up next to each other. Suppose s 1 , s 2 and s 3 are the locations selected with all three distinct.
Combining these formulas with the original conditions yields:. Therefore, positions s 1 ' s 2 ', and s 3 ' are all separated by at least one vacant seat. Therefore, it is invertible. Remove it and you get a unique 8-seat arrangement. Combining these equations with the conditions we have. This is a function that maps a legal seat configuration to a unique 8-seat configuration. The size of a set can be abbreviated s. Using the technique of taking away and adding empty chairs, I can extend the problem to include any number of customers.
You can imagine that three of the ten seats would be introduced by three of the customers. So, there would only be 7 to start with. In general, given n seats and c customers, we remove c- 1 chairs and select the seats for the c customers. Once the number of combinations of seats is found, it is necessary to multiply by c!
Looking at the situation of 3 customers and using a little algebraic manipulation, we get the n P 3 formula shown below. If we multiply n -2 C 3 by 3! After I finished looking at this question as it applied to people sitting in a row of chairs at a counter, I considered the last question, which asked would happen if there were a round table with people sitting, as before, always with at least one chair between them.
I went back to my original idea about each person dragging in an extra chair that she places to her right, barring anyone else from sitting there. There is no end seat, so even the last person needs to bring an extra chair because he might sit to the left of someone who has already been seated. So, if there were 3 people there would be 7 seats for them to choose from and 3 extra chairs that no one would be allowed to sit in.
Given 3 customers and n seats there are n -3 C 3 possible groups of 3 chairs which can be used to seat these customers around a circular table in such a way that no one sits next to anyone else. My first attempt at a proof: To test this conjecture I started by listing the first few numbers generated by my formula:. Then I started to systematically count the first few numbers of groups of possible seats.
I got the numbers shown in the following table. If we remove 3 seats leaving 5 there are 10 ways to select 3 of the 5 remaining chairs. The arrows point to where the person who selects that chair could end up. For example, if chair A is selected, that person will definitely end up in seat 1 at the table with 8 seats. If chair B is selected but chair A is not, then seat 2 will end up occupied. However, if chair A and B are selected, then the person who chose chair B will end up in seat 3.
The arrows show all the possible seats in which a person who chose a particular chair could end. Notice that it is impossible for seat 8 to be occupied.
It does not allow all seats at the table of 8 to be chosen. In a circle, it is not so easy. Using finite differences I was able to find a formula that generates the correct numbers:. I used this formula to predict the next two numbers in the chart and confirmed them by hand. Note that the factored form of this cubic function gives clues to how the problem works. So, we select one and seat person A in that seat. Select another seat and put person B in it.
Finally, take the two seats that were previously removed and put one to the left of B and one to the left of C. This results in the function: In a manner similar to the method I used in the row-of-chairs-at-a-counter problem, this could be proven more rigorously. Consider a grid of chairs in a classroom and a group of 3 very smelly people.
No one wants to sit adjacent to anyone else. There would be 9 empty seats around each person. Suppose there are 16 chairs in a room with 4 rows and 4 columns. How many different ways could 3 people sit? What if there was a room with n rows and n columns? What if it had n rows and m columns? December - February Conversations with my mathematics mentor. Giving an oral presentation about your mathematics research can be very exciting!
You have the opportunity to share what you have learned, answer questions about your project, and engage others in the topic you have been studying. After you finish doing your mathematics research, you may have the opportunity to present your work to a group of people such as your classmates, judges at a science fair or other type of contest, or educators at a conference. With some advance preparation, you can give a thoughtful, engaging talk that will leave your audience informed and excited about what you have done.
In most situations, you will have a time limit of between 10 and 30 minutes in which to give your presentation. Based upon that limit, you must decide what to include in your talk.
Come up with some good examples that will keep your audience engaged. Think about what vocabulary, explanations, and proofs are really necessary in order for people to understand your work. It can be difficult for people to understand a lot of technical language or to follow a long proof during a talk.
As you begin to plan, you may find it helpful to create an outline of the points you want to include. Then you can decide how best to make those points clear to your audience. You must also consider who your audience is and where the presentation will take place. If you are going to give your presentation to a single judge while standing next to your project display, your presentation will be considerably different than if you are going to speak from the stage in an auditorium full of people!
Consider the background of your audience as well. Is this a group of people that knows something about your topic area?
Or, do you need to start with some very basic information in order for people to understand your work? If you can tailor your presentation to your audience, it will be much more satisfying for them and for you.
No matter where you are presenting your speech and for whom, the structure of your presentation is very important. There is an old bit of advice about public speaking that goes something like this: Get the attention of the audience and tell them what you are going to talk about, explain your research, and then following it up with a re-cap in the conclusion.
Your introduction sets the stage for your entire presentation. The first 30 seconds of your speech will either capture the attention of your audience or let them know that a short nap is in order. You want to capture their attention. There are many different ways to start your speech. Some people like to tell a joke, some quote famous people, and others tell stories.
You can use a quote from a famous person that is engaging and relevant to your topic. And yes, I did drink a lot of coffee during the project! Hawking, I am here to convince you that he is wrong. There were 3 regular customers who came in between 6: In fact, these people never sat next to each other. Well, their anti-social behavior led me to wonder, how many different ways could these three grouchy customers sit at the breakfast counter without sitting next to each other?
Amazingly enough, my summer job serving coffee and eggs to grouchy folks in Boston led me to an interesting combinatorics problem that I am going to talk to you about today. It has been said that there are three kinds of mathematicians: All joking aside, my mathematics research project involves counting.
I have spent the past 8 weeks working on a combinatorics problem.. To find quotes to use in introductions and conclusions try: To find some mathematical quotes, consult the Mathematical Quotation Server: There is a collection of math jokes compiled by the Canadian Mathematical Society at http: After you have the attention of your audience, you must introduce your research more formally. You might start with a statement of the problem that you investigated and what lead you to choose that topic.
Then you might say something like this,. In order to do this I will first explain how I came up with this formula and then I will show you how I proved it works.
Finally, I will extend this result to tables with more than 3 people sitting at them. By providing a brief outline of your talk at the beginning and reminding people where you are in the speech while you are talking, you will be more effective in keeping the attention of your audience. It will also make it much easier for you to remember where you are in your speech as you are giving it.
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Writing a Research Paper in Mathematics Ashley Reiter September 12, Section 1: Introduction: Why bother? Good mathematical writing, like good mathematics thinking, is a skill which must be practiced and developed for optimal performance.
Some research papers by Charles Weibel. K-theory of line bundles and smooth varieties (C. Haesemayer and C. Weibel), Proc. AMS (to appear). 11pp. preprint,
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