New material will be added to this file at the top.

**Final Exam and Course Letter Grades Posted**

Here are the final exam and course letter grades posted by the ID number on your final exam paper.

Have a nice summer!

**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).

**Test 4 grades posted**

Here they are, posted by the ID number on your test paper.

**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).

**Test III grades posted**

Here find grades for Test III posted by the ID number on the test paper.

**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 Grades and Solutions**

Here are the grades on Test II posted by the ID number on the test paper. 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 Grades Posted**

Here are the grades from Test I posted by the number on your test paper. 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.

**Homework corrected to fit the new book**

The homework schedule now matches the new book. As the new homework exercises seem to be considerably modified, I will be writing new assignments; there are several posted.

**Welcome**

Im posting this Jan 18, before classes start. Welcome to the class. I have left some material from the last term's announcements page which may be useful. New material on this page will be added at the top. When bits of the legacy material become Useful, they will be copied to the top. This is where to look for links to documents for this class such as worksheets.

If you are trying to get into the section and you do not understand why you need a permission number, you are probably not a math major. This section is for math majors only. You may be able to convince the office staff (not me) that you should be in this section: if you do, be aware that in this section you will be expected to work on reading and writing proofs at a level appropriate for math majors.

**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.