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  • Review sheet for the final is posted, containing some of the definitions and theorems we've covered. This review sheet only covers material since after Exam 3. The final is cumulative.

  • Here is the combinatorial game theory text we'll be following for the rest of the term.

  • Solutions to Exam 3 are here. The median is a 72. Here are the approximate grade cut-offs:
    A: 90+, A-: 84-89, B+: 77-83, B: 70-76, B-: 65-69, C+: 63-64, C: 50-62, C-: 43-49, D: 37-42, R: 0-36

  • Review sheet for Exam 3 is here. Solutions are here.

  • Here is a good probability bookfor extra probability examples and explanations.

  • Clarifications:
    • you can pick up your homework even if you didn't sign the FERPA waiver, and
    • it is not too late to sign the FERPA waiver.
    Reminder: the FERPA waiver grants the ability to pick up homework even when I'm not in my office.

  • Solutions to Exam 2 are here. The mean and median are both around 58 and the approximate grade "cut-off"s are as follows.
    A: 88+, A-: 75-87, B+: 67-74, B: 60-66, B-: 55-59, C+: 52-54, C: 47-51, C-: 45-46, D+: 42-44, D: 38-41, R: 0-37

    If you got a C- or lower, I encourage you to make an appointment to speak with me for 10-15 minutes.

  • Review sheet for Exam 2 is here. Solutions are here.

  • Mini-quizzes: Some people have asked for a list of the questions we've had on mini-quizzes thus far, so here they are.
    1. Compute the Stirling number S(n,n-1).
    2. Consider the sequence {0,1,0,1,...}. Write its generating function in closed form.
    3. Write down the best bounds you know for n! and state whether you can prove these or not.
    4. On average, how many times would you have to flip a pair of coins before seeing both come up heads on the same toss?
    5. Write down an example of an n-vertex graph with chromatic number 2 (for arbitrary n).
    6. 13 ⊕ x = 24. Find x.

  • Solutions to Exam 1 are here. The median is a 50 and the approximate grade cut-offs* are as follows:

    A: 78+, A-: 71-77, B+: 57-70, B: 50-56, B-: 45-49, C+: 40-44, C: 36-39, C-: 34-35, D+: 31-33, D: 25-30, R: 0-24

    *This means that if I had to give you a grade based solely on this exam (ignoring homework, participation, etc.), this is what it would be.

  • More resources and practice problems:
    • Review sheet for Exam 1 is here. Not everything on this review sheet represents something that will be on the exam! It is a sheet that gives the outline of what we've covered up to this point in the class. As pointed out in lecture on 2/11, generating functions will not be a large portion of the material for Exam 1. Here are solutions to the problems on the review sheet. It's recommended that you try the problems before you look at the solutions.

    • Here is an excellent resource for more on the D.I.E. method (includes lots of practice problems as well as good explanations of most of the problems we did in class).

    • Here is a video of a nice, dorky guy explaining the 12-fold way:

    • And here is the same guy explaining inclusion-exclusion:

  • Here is Herb Wilf's book, generatingfunctionology. It's an excellent extra resource for learning about generating functions. I highly recommend you read its first chapter.

  • Combinatorial arguments: Here is a good source to look at if you're not quite sure what is meant by a combinatorial argument. This guide serves as a review of material learned in Concepts.

  • Twelvefold Way: Here is a nice Wikipedia article that outlines the explanations of the twelvefold way entries, which we did in class on 1/23.

  • For more on the correspondence between Towers of Hanoi, the Sierpinski triangle, and Pascal's triangle, see:

  • We saw that Pascal's triangle modulo 2 looks like the Sierpinski triangle. Check out what happens when you look at it mod bigger primes:

    Pascal's triangle mod 3

  • FERPA waiver: The Family Educational Rights and Privacy Act (FERPA) is a Federal law that protects the privacy of student's education records as well as providing a considerable amount of privacy concerning their academic status and grades. As such, graded written homework assignments in this class will not be returned in a public fashion unless you sign a waiver stating that you're OK with it. Whether you sign the waiver or not will have no impact on your performance in this class. If you choose not to sign a FERPA waiver, you will need to visit my office during office hours to pick up your graded homework, and may need to present your student ID. The waiver does not apply to exams, which will be returned in a private fashion. Waivers may be signed at any time after the first day of class. If you choose to sign the waiver, your homework will be returned via envelopes that are hanging on the door of my office, Wean 6130. If you decide to sign the waiver, please print it and hand it in with your first homework assignment.