Name: Exam Score: /100. EXAM 3 Version A. Multiple Choice. (26 questions at 4 points each) Circle the letter of the best answer.

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1 MTH xplorations in Modern Mathematics Fall 2013 ircle one: MW7:45 / MWF1:10 Dr. Kracht Name: xam Score: /100. (104 pts available) XM 3 Version Multiple hoice. (26 questions at 4 points each) ircle the letter of the best answer. cademic Honesty Pledge Your signature at the bottom indicates your agreement to abide by the following rules. 1. ll purses, bags, pencil cases, books, notes, and other papers are placed in the designated area. 2. ll electronic devices (including cell phones, Blackberries, ipods, etc.) except for hand-held calculators are placed in the designated area. 3. I will not share my calculator or Rubik s cube with another student during the exam. 4. I will not communicate with, seek help from, or give help to other students during the exam. 5. I will turn my exam in and will not take it from the classroom. 6. I will not cheat in any other way. Signature:. For problems 1 3: Brenda and ddy buy a 12-inch half-ham, half-turkey sub sandwich for $4.80. (So there are 6 inches of ham and 6 inches of turkey.) ddy values ham three times as much as turkey. 1. What is the value, to ddy, of 1 inch of turkey? (a) $0.60 (b) $0.30 (c) $0.20 (d) $ What is the value, to ddy, of 1 inch of ham? (a) $0.60 (b) $0.30 (c) $0.20 (d) $ Suppose ddy is the Divider in a 2-person Divider-hooser Fair Division Game. He cuts the sub into two pieces crosswise. The two pieces are as follows. (a) One piece consists of 6 inches of ham and the other piece consists of 6 inches of turkey. (b) One piece consists of 4 inches of ham and the other piece consists of 2 inches of ham and 6 inches of turkey. (c) One piece consists of 3 inches of ham and the other piece consists of 3 inches of ham and 6 inches of turkey. (d) One piece consists of 4 inches of turkey and the other piece consists of 2 inches of turkey and 6 inches of ham. 1

2 For problems 4 7: Three students, li, Bobby, and arrie, divide fairly 20 pieces of candy, of different types, using the Method of Markers. The candy is placed in an array and the players bid by placing markers as shown below. (Marker 1 is li s first marker, 2 is her second; B 1 and B 2 are Bobby s markers, and 1 and 2 are arrie s.) It is agreed that candy leftover after the allocation will be kept by the teacher. Recall that our convention is to go from left to right B1 1 B Which pieces does li get? (a) 8 13 (b) 1 7 (c) (d) Which pieces does Bobby get? (a) (b) 7 12 (c) 1 6 (d) Which pieces does arrie get? (a) (b) 1 5 (c) 6 12 (d) Which pieces does the teacher get? (a) 6, 13, and 14 (b) 6 and 13 (c) 6, 7, and 14 (d) 7 and 14 2

3 For problems 8 10: Santa s reindeer agree to divide a $24 pizza fairly using the Lone Divider Method. The table shows how each player values each of the eight slices that have been cut by the divider. ssume that all of Santa s reindeer play honestly. s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 Dasher $5 $2 $4 $2 $2 $5 $2 $2 Dancer $1 $2 $1 $1 $1 $16 $1 $1 Prancer $2 $2 $10 $1 $3 $2 $2 $2 Vixen $10 $2 $2 $2 $2 $2 $2 $2 omet $6 $2 $2 $1 $2 $6 $3 $2 upid $11 $1 $0 $2 $0 $2 $2 $6 Donder $3 $3 $3 $3 $3 $3 $3 $3 Blitzen $4 $4 $4 $2 $4 $1 $1 $4 8. Which reindeer was the divider? (a) Vixen (b) Donder (c) Blitzen (d) It is impossible to tell from the information given 9. What is Dasher s bid list (declaration)? (a) s 1, s 3, s 6 (b) s 1, s 6 (c) s 1, s 2, s 3, s 4, s 5, s 6, s 7, s 8 (d) s Find a fair division of the pizza using slices, if possible. Which piece does Blitzen receive? (a) s 5 (b) s 2 (c) ny one of s 1, s 2, s 3, s 5, or s 8 (d) No fair division is possible using slices. 3

4 For problems 11 13: Four heirs (, B,, and D) must fairly divide an estate consisting of two items, a house and a cabin, using the Method of Sealed Bids. The players bids, in dollars, are shown below. 11. How much is the original fair share of player worth? (a) $240,000 (b) $80,000 (c) $120,000 (d) $60,000 D House 195, , , ,000 abin 45,000 36,000 35,000 42, In the initial allocation, what did player B receive? (a) Player B gets $62,000 from the estate. (b) Player B gets the house and the cabin and pays the estate $186,000. (c) Player B gets the house and pays the estate $274,000. (d) Player B gets the house and pays the estate $150, fter the initial allocation to each player is made, how much is the surplus? (a) $32,000 (b) $0 (c) $20,000 (d) $150,000 4

5 For problems 14 26: Use your cube! ll MOVS are viewed from the FRONT. 14. Rubik s ube has (a) 6 center cubies, 24 edge cubies, and 24 corner cubies. (b) 6 center cubies, 8 edge cubies, and 12 corner cubies. (c) 6 center cubies, 12 edge cubies, and 8 corner cubies. (d) 6 center cubies, 12 edge cubies, and 12 corner cubies. 15. Which of the following is/are possible? (a) to move an edge cubie to a corner position. (b) to move an edge cubie to a center position. (c) to move a center cubie to an edge position. (d) ll of are possible. (e) None of is possible. 16. The move does which of the following? 17. The move does which of the following? 18. The move does which of the following? 5

6 19. The move does which of the following? 20. If Y =, what is Y? (a) (b) (c) (d) D For problems 21 22: onsider. We wish to trace the movement of the corner cubie labeled B and the edge cubie labeled D after successive applications of W =. 21. fter W 2 : (a) (b) (c) (d) (e) None of B B B D D 22. fter W 3 : (a) (b) (c) (d) (e) None of D B B D 6

7 23. If P does Upper without affecting anything else on the Upper Layer, then which of the following does Upper and nothing else? (Recall that U =.) (a) U P U P (b) P U P U (c) P 2 U P 2 U (d) P U 2 P U 2 (e) None of 24. If Q does Up without affecting anything else on the Upper Layer, then which of the following does Up and nothing else? (a) U 2 Q U 2 Q (b) Q U Q U (c) Q 2 U Q 2 U (d) Q U Q U 2 (e) None of For problems 25 26: Suppose Z moves seven edge cubies. Some of edge cubies are back to their original positions and orientations after 2 applications of Z, while some require 5 applications of Z to return to their original positions and orientations. 25. What is the least number of applications of Z required to return all of the moved edge cubies to their original positions and orientations? (a) 5 (b) 10 (c) 7 (d) Suppose Z also moves six corner cubies. ll of corner cubies require 6 applications of Z to return to their original positions and orientations. What is the least number of applications of Z required to return all of the moved cubies (both edges and corners) to their original positions and orientations? (a) 6 (b) 13 (c) 30 (d) 60 The following questions are for information only. They will not count toward your grade. Did you solve (at least) the Upper Layer? Yes No Did you solve (at least) the Upper and Middle Layers? Yes No Did you solve the whole cube? Yes No Did you enjoy the Rubik s ube unit? Yes No If you still have not solved your whole cube but would like to, I ll be happy to continue to help you to work on it after the exam! 7