Download Math Exam 3 for Math 181 X1: Stable Marriage Problem, Game Theory, Ciphers and more Exams Mathematics in PDF only on Docsity!
Math 181 X1 Exam 3 Student’s Name: Instructor: Rosona Eldred Date: 4/25/
This exam has 9 questions, for a total of 100 points
- You will have 50 minutes to complete the exam. I will try to give you 5 minutes after the bell rings as well. At that point,I will collect the exams without exception.
- Points are indicated in each question.
- If you need extra space, use the extra pages at the back of the exam.
- CLOSED BOOK, CLOSED NOTES, CLOSED NEIGHBOR. No calculators!
- No cell phones/mp3 players/PEDs
- REMEMBER THAT CHEATING IS TAKEN SERIOUSLY AND MAY RESULT IN YOUR DIS- MISSAL FROM THE UNIVERSITY.
- Good luck.
Page Points Score 2 8
3 26 4 16 5 26
6 12 7 12 Total: 100
- (8 points) Stable Marriage Problem 12 of your close friends come to you and tell you that they’re hopelessly in trouble: no matter how they pair themselves up and start dating, someone inevitably ends up cheating on someone else! They beg you to pair them up so that they are in stable relationships: no two of them are going to cheat with each other on their partners. Find a stable marriage for all of your friends, using the algorithm as in class. You need to have a column for proposals as well as for engagements, crossing off engagements when they become voided. I started the chart for you. Please expand it and fill it out (e.g. you’ll probably have more than 2 rounds). Assume women propose to men
Women Ordered Preferences to Marry Ann Kyle > Han > Ib > Jan > Liam > Greg Bryn Greg > Han > Ib > Liam > Kyle > Jan Cher Jan > Greg > Liam > Han > Ib > Kyle Dara Greg > Ib > Jan > Han > Kyle > Liam Ella Jan > Greg > Ib > Han > Kyle > Liam Fran Greg > Jan > Liam > Kyle > Han > Ib Men Ordered Preferences to Marry Greg Bryn > Cher > Dara > Ann > Ella > Fran Han Fran > Bryn > Ann > Cher > Dar > Ella Ib Ella > Bryn > Dara > Fran > Ella > Cher Jan Bryn > Cher > Dara > Ann > Fran > Ella Kyle Bryn > Fran > Ann > Cher > Dara > Ella Liam Ella > Ann > Bryn > Dara > Fran > Cher
Proposals Engagements Round 1
Round 2
- Given the following payoff matrix
Column Player C1 C2 C Row Player R1 3 -8.9 1. R2 1 6.1 5 (a) We want to determine Row Player’s optimal mixed strategy i. (2 points) (circle one) R, the matrix of Row Player’s mixed strategies, is a (2x1/1x2) matrix ii. (4 points) Fill in the equations of lines for Cloumn player choosing C1, C2 or C3: equation at x = 0 at x = 1 Line for C 1 Line for C 2 Line for C 3
iii. (4 points) Graph the lines. LABEL the points of intersection, and label each line with the choice on Column Player’s part that it corresponds to (e.g. C 1 )
iv. (2 points) What is Row Player’s optimal mixed strategy?
(b) We want to determine Column Player’s optimal mixed strategy, knowing Row Player’s. i. (2 points) (circle one) C, the matrix of Column Player’s mixed strategies, is a (2x1/1x2) matrix
ii. (2 points) Using your work from the figuring out Row Player’s optimal mixed strategies, which strategy should Column Player not play? (this is like when I asked which two of the three strategies should Column Player use, except I’m asking you to specify which one should be omitted).
- Using the cipher C ≡ 11 ρ + 3 mod 26
(a) (5 points) Encode ”brillig”
(b) (2 points) What is my decoding recipe for this cipher?
(c) (4 points) Decode RHIV.
- Using the cipher C 1 ≡ 3 P 1 + 10P 2 mod 26 C 2 ≡ 2 P 1 + 15P 2 mod 26 (a) (4 points) Encode HI
(b) (2 points) What is the inverse of ad − bc mod 26?
(c) (5 points) What is my decoding recipe for this cipher?
P 1 ≡ mod 26 P 2 ≡ mod 26
(d) (4 points) Decode XC
- The following was encoded with a shift cipher:
Gsjsfoz ct hvs kcfrg wb hvs dcsa ofs ct Qoffczz’g ckb wbjsbhwcb, aobm ct hvsa dcfhaobhsoil. Wb hvs pccy, hvs qvofoqhsf ct Viadhm Riadhm uwjsg rstwbwhwcbg tcf hvs bcbgsbgs kcfrg wb hvs twfgh ghobno. Wb zohsf kfwhwbug, Zskwg Qoffczz sldzowbsr gsjsfoz ct hvs chvsfg. Hvs fsgh ct hvs bcbgsbgs kcfrg ksfs bsjsf sldzwqwhzm rstwbsr pm Qoffczz, kvc sjsb qzowasr hvoh vs rwr bch ybck kvoh gcas ct hvsa asobh. Ob slhsbrsr obozmgwg ct hvs dcsa wg uwjsb wb hvs pccy Hvs Obbchohsr Ozwqs, wbqzirwbu kfwhwbug tfca Qoffczz opcih vck vs tcfasr gcas ct vwg wrwcgmbqfohwq kcfrg. O tsk kcfrg hvoh Qoffczz wbjsbhsr wb hvwg dcsa (boaszm ”qvcfhzsr”, ”uoziadvwbu”, ”tfopxcig”, obr ”jcfdoz”) vojs sbhsfsr hvs Sbuzwgv zobuious. Hvs kcfr xoppsfkcqym whgszt wg gcashwasg igsr hc fstsf hc bcbgsbgs zobuious. (a) (6 points) Decode the (last two) words, ” bcbgsbgs zobuious ”
(b) (2 points) What was the original cipher used to encode the message? (i.e. write out C ≡ ρ + b mod 26 and replace b with the shift that was actually used)
(c) (4 points) Explain your method for (a).
