CSE 21 Homework 0 Due: Friday October 8, 2017 at 11:59pm

Instructions This homework assignment must be completed individually (not in groups). Completing this assignment will set you up to use the tools you’ll need for the rest of the course, and will give you and us an idea of your mastery of the prerequisite knowledge before you start the course. Required Reading Rosen Chapters 1 and 2, Sections 5.1-5.2 (covers prerequisite concepts).

1. Make sure you can access the Gradescope site for this class. • Navigate to http://gradescope.com. • Login with your @ucsd email account. • Having trouble? You can manually add this course using entry code: MD532D. Please use your official full name when signing up. 2. Enroll in Piazza. • Use the signup link: http://piazza.com/ucsd/sprinng2016/cse21bd. 3. Register your iClicker remote. • Use https://tritoned.ucsd.edu. • Navigate to our class and look for “Register your iClicker” in the left side bar.

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1. (10 points) Prove the following fact about convex polygons using induction: The sum of the interior angles of any convex n-gon is 180 ∗ (n − 2). (You may assume without proof that the interior angles of any triangle sum to 180 degrees.) Please write out the • Base Case. • Inductive Hypothesis. • What you want to show. • The Inductive step. (Your argument should be clear and concise.) 2. (10 points) Consider the following algorithm procedure Loops(n: a positive integer) 1.

for i := 1 to n

2.

for j := i to n

3.

print (i, j)

(a) Write what the algorithm prints when n = 4. (b) Describe what the algorithm prints in general terms. (c) How many times does print routine get called? (d) Describe (in words) a rule to decide, if (i1 , j1 ) and (i2 , j2 ) have both been printed for some n then which ordered pair was printed first? 3. (10 points) Find the flaw with the following “proof” that every postage of three cents or more can be formed using just 3-cent and 4-cent stamps. Base Cases: We can form postage of three cents with a 3-cent stamp and four cents with a 4-cent stamp. inductive step: Assume that for some k > 4 we can form postage of j cents for all nonnegative integers j with 3 ≤ j ≤ k using just 3-cent and 4-cent stamps. We can then form a postage of k + 1 cents by replacing one 3-cent stamp with a 4-cent stamp or by replacing two 4-cent stamps with three 3-cent stamps. 4. (10 points) Use strong induction to show that every positive integer n can be written as a sum of distinct powers of two. (Hint: for the inductive step, separately consider the case where k + 1 is even and where it is odd. When it is even, note that (k + 1)/2 is an integer.)

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