New material will be added to this file at the top. The homework schedule has been moved to the syllabus.

**Test 4, Final Exam and Course Grades**

Here are the Test 4, Final Exam and course grades.

**Final Exam Resources**

The hour exams are of course study resources for this test. Here is a set of solutions to your Test I. Here is Test II with solutions. Here is Test III with solutions. All of the old hour exams are study resources as well.

Here is the graph theory worksheet with solutions. Here are detailed solutions to two of the problems on the RSA worksheet.

Here is the Spring 2013 final exam.

Here is the Spring 2012 final exam. You will not have a proof question about graph theory.

Comments on the following final papers were written in Spring 2012. Their coverage was not identical to yours but they should be useful.

M187Su06final.pdf A good study resource for you – everything here is something I might ask. This particular test has some irritating typos in it.

M187F06final.pdf we didnt do bubble sort; otherwise this is quite similar in coverage.

M187S07final.pdf This is a nice study resource for you, too. Since it is a summer exam, it has quite a lot of the later material (it is also test 4, as it were).

M187F08final.pdf Disregard the question about groups. This was from the semester where we were trying to save paper and made the students use blue books, so it doesn't have a separate page for each question.

M187S10final.pdf Also a good test. Notice that problems in the same area are not always phrased the same in these exams; these tests are not clones (some questions are pretty stereotyped, of course).

**Graph Theory Problems**

Here is the last homework set. You can turn it in at the final; I will post solutions for it before the final.

**Review Sessions Are Scheduled**

10—11:30 AM on the 16^{th} and 17^{th} Dec
(Monday and Tuesday before our final on Wednesday) I will hold
additional review sessions in B102B (Business Building). This is
entirely optional, but gives you an extra opportunity to ask
questions and work on problems.

I'll share with you the warning the scheduling people gave me: if a final shows up in the room we are scheduled for, or if one comes in while we are there, we have to leave. They say they don't expect any problems, but if there are, people taking finals win.

**RSA examples and worksheet**

Here is a small worked example of the RSA algorithm, with explanations. Here is your homework sheet; notice that the numerical answers are given – your aim is to find the work that gives these answers. Do note that you are expected to factor the values of N given into primes yourself.

**Test III Grades Posted**

Here are the Test III grades. Question 9 was treated as extra credit, the lowest other question on your paper was reduced in weight by half.

**Modular Exponentiation Worksheet**

Here is a worksheet on the topics covered in Friday's lecture. It is due Friday after break.

**Test III Resources**

Here are solutions to Test 3 from the last time I taught the class. Sample Test III papers: Fall 2006; Fall 2008; Spring 2007; Summer 2006; Summer 2007. Spring 2012.

The sections covered on our Test III are 18, 19 (earlier counting principles might be used), 21/22 (no requirement to use minimal counterexample), 23, 24, 35, 36. You may find relevant questions on some Test 2 or Test 4 papers given below, and of course you might find relevant papers on old finals given below.

Here find my spreadsheet Euclidean algorithm calculator.

**Important Announcements, Please Read (includes
Test Dates)**

The induction assignment
is due Friday the 8^{th}, Monday was a typo.

Test III will be on Wednesday November 20^{th}. Test IV
will be folded in with the cumulative final, and will happen in the
final exam period on Dec 18th. The test you take on the 18^{th}
will be divided into Test IV questions and a cumulative part, with
some modifications in the overall grade formula. I have laid out the
schedule of sections to be covered for the rest of the course.

It appears possible that I will be absent for professional reasons
on December 13^{th} (the last day of classes). Notice that
Dec 11 is labelled review for final. If this happens, I will cancel
class (I do not see the point of having a colleague cover the review
for my final) but will offer review sessions on Monday and Tuesday of
the following week.

**Induction Assignment, remark about Monday 9 am
office hour**

Sorry about zoning out about the induction assignment! I have just posted it (Monday morning); it covers things I will talk about on Wednesday as well as what I talked about Friday, and will be due the following Wednesday. Get started on it. Notice that in section 21, you are invited to use induction rather than least counterexample. You should be ready to attempt 21 3,4; 22.4 all parts; 22.5 ab (be very careful with c!). Careful with inequality reasoning in general. That is not to say you might not have success with some of the others, but there are additional issues which I will talk about on Wednesday.

**About the Monday 9-10:15 office hour:** I am holding it! I am
sometimes late, and the fact that I am holding it is obscured by the
fact that a stats instructor is holding a large study session there
at the same time; it looks as if there is a class there, but the lab
is actually open.

**Last Counting Assignment**

Here is a sheet of counting problems. Do problem 5,10,11,12. Here is another. Do problems 4,5,11,12. The book doesn't supply word problems for either section. Also do problem 20 in the section 3 self test.

All of these problems would be Good for You.

**Test II Grades Posted**

Here are the Test II grades posted by the ID number on your paper.

**Test II on Friday Oct 18; Review Materials Moved
Up.**

Sample Test II papers: Fall 2006 Test II; Fall 08 Test II; Spring 2007 Test II; Spring '10 Test II; Summer 06 Test II; an old Test II review sheet Spring 2012 Test II paper. Here are the solutions to the Spring 2013 Test II. (I just moved this up here, but in fact it has appeared below in this page from the beginning of the class; all review material which will be available for each exam except the final was already posted when class started).

**Notes I Promised for a Proof done in Class**

Here are the notes for the theorem about symmetric difference that I proved in class.

**Computer Lab Information**

Here is the lab manual. This needs to be updated, as I will explain at lab time, but I don't have access to the source right now. We should be able to work with it; I will update it when I can. I may also add some other links here. The lab assignment is due a week after we meet in the lab (and I really encourage you to get it in by then) but it is worth noting for this and future lab assignments that I accept labs late without penalty; this may seriously mess up how soon I get to grading them, though.

**Test I Grades and Solutions**

Here are the Test I grades posted by the ID number on your test paper. Here is a set of solutions to the test.

**Additional Office Hour for M187**

9 am – 10:15 am Monday in MG136 (the new Math computer lab,
a good place). I'm here as I write this on the 16^{th} at
9:36 am. Note that this hour will not be held in M136 but in my
office if the lab has been reserved for a class, which will probably
happen now and then.

**Office Hours**

10:30—11:30 AM MWF and 3—4PM WF. I have logic seminar at 3 PM Mondays.

**Formal Logic Materials**

Here is our manual of logical style.

**Homework Schedule**

The homework schedule now appears in the syllabus on my main page.

**Welcome to the Class**

For the moment, this is a clone of the last version of the Spring 2013 announcements. I will edit this to include only stuff likely to be useful to us eventually.

**Old Stuff from SP 2013**

**Test 3 and 4 Solutions**

Here are solutions to Test 3. Here are solutions to Test 4. Happy studying!

**Sample final exam papers**

Here is the Spring 2012 final exam. It differs from yours in coverage only by having a proof question about graph theory.

Comments on the following final papers were written in Spring 2012. Their coverage was not identical to yours but quite similar.

M187Su06final.pdf A good study resource for you – everything here is something I might ask. This particular test has some irritating typos in it.

M187F06final.pdf we didnt do bubble sort; otherwise this is quite similar in coverage.

M187S07final.pdf This is a nice study resource for you, too. Since it is a summer exam, it has quite a lot of the later material (it is also test 4, as it were).

M187F08final.pdf Disregard the question about groups. This was from the semester where we were trying to save paper and made the students use blue books, so it doesn't have a separate page for each question.

M187S10final.pdf Also a good test. Notice that problems in the same area are not always phrased the same in these exams; these tests are not clones (some questions are pretty stereotyped, of course).

**Sample Test 4 papers**

Here are sample Test 4 papers. These may (will) differ substantially in coverage from our Test IV, which is entirely on number theory. You should also look at the Test III papers for number theory problems. We are covering sections 35-39, 43 and 46 in one way or another. I'll talk more about coverage on Wednesday. Fall 2006; Spring 2007; Summer 2006 final (this has a lot of Test 4 questions). The Spring 2012 Test 4 is here.

**Worksheet on RSA encryption**

Here is the worksheet. Here is a fully worked tiny example (with extremely small numbers).

**Worksheet on modular exponentiation**

Here it is! This is a worksheet on exponentiation in modular arithmetic, which you can do by the repeated squaring method demonstrated in class (I'll do another example on Wed) and (when allowed) by the use of Fermat's little theorem a^(p-1) = 1 mod p (for p a prime).

**Sample Test III Papers**

Sample Test III papers (sorry these are late): Fall 2006; Fall 2008; Spring 2007; Summer 2006; Summer 2007. Here is the most recent paper: Spring 2012. Happy studying! I will put up a study guide here later on Friday (sections and comments on what I am likely to ask).

The headings for this test are: counting problems (I could ask some of these); math induction proofs: there will be two of them, one a summation problem and one not; functions and composition (look at what I asked in homework; I am interesting in counting functions so review the theorem about counting functions from a set with m elements to a set with n elements, and counting one-to-one functions from a set with m elements to a set with n elements). On functions and compositions you should of course take a look at what homework problems I marked to get an idea of what a routine problem would look like. I will ask questions about permutations (section 27). I like problem 27.13; make sure you understand how to answer it. Make sure you know how to classify a permutation as odd or even (composition of an odd or even number of transpositions). I will ask a Pigeonhole Principle problem (look at examples on the tests): a large part of what I will be looking for is an understanding of what the Pigeonhole Principle says. I might ask a problem about surprising one-to-one correspondences between infinite sets; I will be looking for an understanding of patterns rather than explicit formulas in such a question, if I ask one.

**Test II Solutions**

Here are the solutions.

**Counting Problems**

Here is a little sheet of counting problems. They will be due some time after the test. Some of them are good practice for the test (the simpler ones!)

**Sample Test II papers**

Happy Studying. I may add some comments about which problems to study later.

Sample Test II papers: Fall 2006 Test II; Fall 08 Test II; Spring 2007 Test II; Spring '10 Test II; Summer 06 Test II; an old Test II review sheet Spring 2012 Test II paper

**Test I Solutions**

Here is my solution set.

**Remark about quizzes**

I very occasionally give quizzes in class. I was thinking of giving one today and very likely will give it on Friday. A quiz is marked as a homework assignment and there are no makeups; if you happen to miss one the saving grace is that I drop a certain number of homework assignments at the end of the term. I'll be happy to give you a copy of a quiz to work on if you missed it.

**Test Date Changed**

The test is Feb 22 not Feb 15. I have revised the schedule to reflect accurately what I plan to cover before the test. Material covered by the end of class Monday before the Friday test would normally be fair game for the test, but of course the Monday is a holiday, so in this case material covered by the end of class Friday is fair game. It looks as if the coverage on this test will be identical to that on the 2012 test, except that you might have a formal proof in the style of the second logic worksheet. You will also have more time to do the test than they did.

Samples of Test I papers from previous terms: Spring 2010 Test I Spring 2007 Test I Fall 2008 Test I

Fall 2006 Test I. Here is the Spring 2012 Test I paper.

A universal remark about posted old tests is that I draw pictures on tests by hand, and these will not appear in the PDF versions of the tests that I have. If you want my opinion about what a missing diagram would have looked like, ask me.

**Formal Logic Materials**

Here is our manual of logical style.

Here is your second logic worksheet. Use the manual of logical style above for reference; it contains descriptions of the rules.

**Legacy stuff from the Spring 2012 announcements
page that might be Useful**

Sample final exam papers from past semesters:

Here is the answer key for Test 4.

Here are the Test 4 grades.

Here is a handwritten sheet with some graph theory problems useful for the test. more graph theory

Here are sample Test 4 papers. These may differ substantially in coverage from our Test IV. We are covering sections 37, 38, 42, 45, 24, 46, 48, 50 in one way or another. I'll talk more about coverage on Wednesday. Fall 2006; Spring 2007; Summer 2006 final (this has a lot of Test 4 questions); look at problem 7 on in the summer 2006 final.

Here is the section 45 worksheet (two extra problems).

Here are the grades on Test III. You will also get an estimate of your standing in the course on your test paper when it is returned.

Here is the section 42 worksheet. Notice that there are also section 38 problems assigned Friday on the schedule now.

Sample Test III papers (sorry these are late): Fall 2006; Fall 2008; Spring 2007; Summer 2006; Summer 2007. Happy studying!

Definition: an integer x is *special* if and only if there is
a unique integer y such that 1<y<x and y|x.