Abstract Rules: Rule 1. O 7, 8 or 9 Rule 2. XPR or RPX Rule 3. N R Rule 4. N(S), (S)N, N(U), (U)N

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1 Questions 1 5 Each of seven travelers Norris, Oribe, Paulsen, Rosen, Semonelli, Tan, and Underwood will be assigned to exactly one of nine airplane seats. The seats are numbered from 1 through 9 and arranged in rows as follows: Front row: Middle row: Last row: Only seats in the same row as each other are immediately beside each other. Seat assignments must meet the following conditions: Oribe s seat is in the last row. Paulsen s seat is immediately beside Rosen s seat and also immediately beside an unassigned seat. Rosen s seat is in the row immediately behind the row in which Norris seat is located. Neither Semonelli nor Underwood is seated immediately beside Norris. Abstract Rules: Rule 1. O 7, 8 or 9 Rule 2. XPR or RPX Rule 3. N R Rule 4. N(S), (S)N, N(U), (U)N Potential inferences: Inf 1. Row 1: [N somewhere] Because O is in last row, N is in front of R, R is in a full row so can t be in last row with O Inf 2. Row 2: [XPR or RPX] See above information Row 3: [O somewhere] See Rule 1 1. Which one of the following is a pair of travelers who could be assigned to seats 2 and 8, respectively? (A) Norris, Semonelli (B) Oribe, Underwood (C) Paulsen, Oribe (D) Rosen, Semonelli (E) Underwood, Tan a. CORRECT ANSWER e.g., row 1: xnt; row 2: xpr; row 3: USO b. O can t be in row 1 (which is where seat 2 is) see rule 1 c. P can t be in row 1 (which is where seat 2 is) see inf 1 d. R can t be in row 1 (which is where seat 2 is) see inf 1 e. U can t be in row 1 (which is where seat 2 is) see inf 1 to notice N is in the first row. If N is there U can t be in the middle of the row or would have NU or UN which can t be Rule If Semonelli and Underwood are not assigned to seats in the same row as each other, which one of the following must be false? (A) Norris is assigned to seat 2. (B) Paulsen is assigned to seat 5. (C) Rosen is assigned to seat 4. (D) Tan is assigned to seat 2. (E) Underwood is assigned to seat 1. Inf new: Pair 1: row 1: [S and N] and row 3: [U and O] or Pair 2: [U and N]; row 3: [S and O] A. CORRECT ANSWER. N can t be in seat 2 if either S or U is in row 1 see inf new that either U or S is in row 1 with N but N can t be in middle of row or would have NU or UN which can t be Rule 4. So you found the one that must be false so you can stop! 3. If Semonelli is assigned to a seat in the same row as Underwood, which one of the following travelers could be assigned to a seat immediately beside one of the unassigned seats? (A) Oribe (B) Rosen (C) Semonelli (D) Tan (E) Underwood

2 IF S and U both in same row it can t be Row 1 (otherwise violates Rule 4) so must be Row 3 leading to New Inf 1. Row 3: [some combo of USO] Question asks for who could be next to unassigned seat (X)? (A) Can t be O since O s row is already full (see New Inf 1) Row 3: [some combo of USO] (B) Can t be R since Inf 2 tell us Row 2 is either XPR or RPX so R is an asile and next to P (C) Can t be S since S s row is already full (see New Inf 1) Row 3: [some combo of USO] (D) CORRECT ANSWER e.g., row 1: NTx or TxN (E) Can t be U since U s row is already full (see New Inf 1) Row 3: [some combo of USO] 4. If the seat to which Tan is assigned is immediately beside a seat assigned to another traveler and also immediately beside one of the unassigned seats, which one of the following must be true? (A) Tan is assigned to a seat in the front row. (B) Tan is assigned to a seat in the last row. (C) Oribe is assigned to a seat immediately beside Semonelli. (D) Oribe is assigned to a seat immediately beside Tan. (E) Semonelli is assigned to a seat immediately beside Underwood. If xt? or?tx, then that? can t be in Row 3 with O or both U and S would have to be in Row 1 with N which is not possible. Therefore the T has to be in Row 1 and the? has to be N: Row 1: xtn or NTx (A) CORRECT ANSWER (B) No because US can t both be in Row 1. [USN] combo violates Rule 4 (C) No. This is a can be not a MUST: Row 3 could be OUS so not a MUST (D) No. O and T have to be in different rows (E) No. This is a can be not a MUST: Row 3 could be SOU so not a MUST 5. If Oribe is assigned to a seat immediately beside one of the unassigned seats, which one of the following must be true? (A) Oribe is assigned to seat 8. (B) Tan is assigned to seat 2. (C) Underwood is assigned to seat 1. (D) Seat 4 is unassigned. (E) Seat 9 is unassigned. If?Ox, Ox?,?xO, or xo? then T has to be in Row 1 because U and S can t be both in Row 1. Row 1 could be STN, NTS, UTN, or NTU. New inf: Row 1 has T in seat 2. (A) No. Row 3 could be?uo so O could be in seat 9. This is a could not a MUST. (B) CORRECT ANSWER (see new inf. T is a buffer) (C) No. Row 1 could be STN so S could be in seat 1. This is a could not a MUST. (D) No. Row 2 could be RPX so R could be in seat 4. This is a could not a MUST. (E) No. Row 3 could be?uo so O could be in seat 9. This is a could not a MUST. Questions 6 12 A university library budget committee must reduce exactly five of eight areas of expenditure G, L, M, N, P, R, S, and W in accordance with the following conditions: If both G and S are reduced, W is also reduced. If N is reduced, neither R nor S is reduced. If P is reduced, L is not reduced. Of the three areas L, M, and R, exactly two are reduced. Abstract Rules: Rule 1. IF GS GSW Rule 2. IF N (R)(S) Rule 3. IF P (L) Rule 4. LM(R) or LR(M) or RM(L) 6. Which one of the following could be a complete and accurate list of the areas of expenditure reduced by the committee? (A) G, L, M, N, W (B) G, L, M, P, W (C) G, M, N, R, W (D) G, M, P, R, S (E) L, M, R, S, W (A) CORRECT ANSWER

3 (B) Violates Rule 3 (can t have both P and L) (C) Violates Rule 2 (can t have both N and R) (D) Violates Rule 1 (G and S, therefore also need W) (E) Violates Rule 4 (can t have L, M, and R; only 2 of those) 7. If W is reduced, which one of the following could be a complete and accurate list of the four other areas of expenditure to be reduced? (A) G, M, P, S (B) L, M, N, R (C) L, M, P, S (D) M, N, P, S (E) M, P, R, S (A) Violates Rule 4 (need either L or R in addition to M) (B) Violates Rule 2 (can t have both N and R) (C) Violates Rule 3 (can t have both P and L) (D) Violates Rule 2 (can t have both N and S) (E) CORRECT ANSWER 8. If P is reduced, which one of the following is a pair of areas of expenditure both of which must be reduced? (A) G, M (B) M, R (C) N, R (D) R, S (E) S, W (B) CORRECT ANSWER Since P is reduce, we know that L is not (Rule 3) and and Rule 4 says M and R if not L 9. If both L and S are reduced, which one of the following could be a pair of areas of expenditure both of which are reduced? (A) G, M (B) G, P (C) N, R (D) N, W (E) P, S (A) CORRECT ANSWER (B) Violates Rule 3 (can t have both P and L) (C) Violates Rule 2 (can t have both N and R) (D) Violates Rule 2 (can t have both N and S) (E) Violates Rule 3 (can t have both P and L) 10. If R is not reduced, which one of the following must be true? (A) G is reduced. (B) N is not reduced. (C) P is reduced. (D) S is reduced. (E) W is not reduced. TRICK HERE IS TO IGNORE COULD S AND ONLY LOOK FOR MUST S. Notice sometimes you have to try out the answers. These take longer to solve. Reduce Not reduce R Rule 4: LM R Rule 3: LM RP (A) CORRECT ANSWER LMNSW RPG (violates Rule 2 is you try to reduce G so that is that answer. G must be reduced) (B) Notice reducing N is possible. LMGSW RPN (C) Can t have both L and P (Rule 3) (D) Notice reducing S is possible. LMGNW RPS (E) Can t have both N and S (Rule 2)

4 11. If both M and R are reduced, which one of the following is a pair of areas neither of which could be reduced? (A) G, L (B) G, N (C) L, N (D) L, P (E) P, S Reduce Not reduce MR Rule 4: MR L Rule 2: MR LN (A) G, L Check if G can be reduced: MRGSW LNP (no violation) (B) G, N Check if G can be reduced: MRGSW LNP (no violation) (C) CORRECT ANSWER (see rules 4 and 2 above, neither L nor N can be reduced!) (D) L, P Check if P can be reduced: MRPSW LNG (no violation) (E) P, S Check if P can be reduced: MRPSW LNG (no violation) 12. Which one of the following areas must be reduced? (A) G (B) L (C) N (D) P (E) W Reduce Not reduce Option 1: LM RP Option 2: LR MNP Option 3: RM LN (A) G Now try to reduce G: LMNSW RPG (violates Rule 2) (B) L See Option 3 (C) N See Option 3 (D) P See Option 2 (E) W LR MNP (no room for G) RMPSW LNG (no violation) CORRECT ANSWER Can t reduce W in any of these cases: LMNSG RPW (violates Rule 2) LR MNP (no room for W) RMPSG LNW (violates Rule 1) Questions A jeweler makes a single strand of beads by threading onto a string in a single direction from a clasp a series of solid colored beads. Each bead is either green, orange, purple, red, or yellow. The resulting strand satisfies the following specifications: If a purple bead is adjacent to a yellow bead, any bead that immediately follows and any bead that immediately precedes that pair must be red. Any pair of beads adjacent to each other that are the same color as each other must be green. No orange bead can be adjacent to any red bead. Any portion of the strand containing eight consecutive beads must include at least one bead of each color. Abstract Rules: Rule 1. PY RPYR or YP RYPR Rule 2. Pairs GG (OO) (RR) (YY) (PP) Rule 3. (RO) or (OR) Rule [GORPY somewhere] Potential inferences: Inf 1: (O)RPYR(O) Inf 2: (R)RPYR(R)

5 13. If the strand has exactly eight beads, which one of the following is an acceptable order, starting from the clasp for the eight beads? (A) green, red, purple, yellow, red, orange, green, purple (B) orange, yellow, red, red, yellow, purple, red, green (C) purple, yellow, red, green, green, orange, yellow, orange (D) red, orange, red, yellow, purple, green, yellow, green (E) red, yellow, purple, red, green, red, green, green New Inf 1: [GORPY somewhere] (A) No. Can t have RO (Rule 3) (B) No. Can t have RR (Rule 2) (C) CORRECT ANSWER (D) No. Needs the R following RYP (Rule 1) (E) No. Needs O (Rule 4) 14. If an orange bead is the fourth bead from the clasp, which one of the following is a pair that could be the second and third beads, respectively? (A) green, orange (B) green, red (C) purple, purple (D) yellow, green (E) yellow, purple ?? O New Inf 1: (R) O (Rule 3) New Inf 2: (O) O (Rule 2) New Inf 3: (Y or P) (Y or P) O (Rule 1) (A) No. Can t be OO. See New Inf 2. (B) No. Can t be RO. See New Inf 1. (C) No. Can t be PP. See Rule 2. (D) CORRECT ANSWER (E) No. Can t be YPO. See New Inf If on an eight-bead strand the second, third, and fourth beads from the clasp are red, green, and yellow, respectively, and the sixth and seventh beads are purple and red, respectively, then which one of the following must be true? (A) The first bead is purple. (B) The fifth bead is green. (C) The fifth bead is orange. (D) The eighth bead is orange. (E) The eighth bead is yellow. New Inf 1: Need O somewhere (Rule 4) New Inf 2: R G Y P R New Inf 3: (Rule 2) (Y) New Inf 4: (Rule 2) (P) New Inf 5: (Rule 2) (R) (R) New Inf 5: (Rules 3 and 4) (O) O (O) (C) CORRECT ANSWER. See New Inf 5. O has to be somewhere (Rule 4) but it can t be 1st or last because of (OR) rule (Rule 3). 16. If on a six-bead strand the first and second beads from the clasp are purple and yellow, respectively, then the fifth and sixth beads CANNOT be (A) green and orange, respectively (B) orange and green, respectively (C) orange and yellow, respectively (D) purple and orange, respectively (E) yellow and purple, respectively New Inf 1: P Y (?) (?) New Inf 2: (Rule 1) R New Inf 3: (Rule 2) (R) New Inf 4: (New Inf 3 & Rule 1) P Y New Inf 5: (New Inf 3 & Rule 1) Y P

6 (E) CORRECT ANSWER. Can t be YP because you would need R in front of YP and can t have RYP if you already have PYRRYP because you would have RR. 17. If on a nine-bead strand the first and fourth beads from the clasp are purple, and the second and fifth beads are yellow, which one of the following could be true? (A) The seventh bead is orange. (B) The eighth bead is green. (C) The eighth bead is red. (D) The ninth bead is red. (E) The ninth bead is yellow. New Inf 1: Need ROG somewhere in 1st 8 (Rule 4) P Y P Y New Inf 2: (Rule 1) R R New Inf 3: (Rule 3) O New Inf 4: G G or G P Y R P Y R G O Y (A) No. O can t be 7th because can t have RO. Rule 3. (B) No. O has to be 8th because you can t RO (Rule 3) and need O in 1st 8. (C) No. R can t be 8th because O has to be 8th because you can t RO (Rule 3) and need O in 1st 8. (D) No. R can t be 9th because since O is 8th you can t have OR (Rule 3) and need O in 1st 8. (E) CORRECT ANSWER 18. If on an eight-bead strand the first, second, third, and fourth beads from the clasp are red, yellow, green, and red, respectively, then the fifth and sixth beads CANNOT be (A) green and orange, respectively (B) green and purple, respectively (C) purple and orange, respectively (D) purple and yellow, respectively (E) yellow and orange, respectively New Inf 1: Need O somewhere (Rule 4) New Inf 2: R Y G R (?) (?) (D) CORRECT ANSWER. Can t be PY because 7th would need to be R (Rule 1) leaving only 1 space. But because of Rule 4 you still need O but you can t have RO (Rule 3) so will not work. Questions At an evening concert, a total of six songs O, P, T, X, Y, and Z will be performed by three vocalists George, Helen, and Leslie. The songs will be sung consecutively as solos, and each will be performed exactly once. The following constraints govern the composition of the concert program: Y must be performed earlier than T and earlier than O. P must be performed earlier than Z and later than O. George can perform only X, Y, and Z. Helen can perform only T, P, and X. Leslie can perform only O, P, and X. The vocalist who performs first must be different from the vocalist who performs last. Rule 1. Y {} T and Y {} O Rule 2. O {} P {} Z Rule 3. G: X, Y, Z Rule 4. H: T, P, X Rule 5. L: O, P, X Rule 6. 1st not = 6th [ T ] Inf 1. Y {} O {} P {} Z Inf 2. G: YZ H: T and (P or X) L: O and (P or X) Because if G: X then H: TP and L: O and nothing left Inf 3. Since G: YZ these can t be 1st and 6th

7 19. Which one of the following is an acceptable schedule for the performance of the songs, in order from the first to last song performed? (A) X, T, Y, O, P, Z (B) X, Z, Y, T, O, P (C) Y, O, P, X, T, Z (D) Y, P, O, Z, T, X (E) Y, X, O, P, Z, T (A) Violates Rule 1 (B) Violates Rule 2 (C) Violates Inf 3 (D) Violates Rule 2 (E) CORRECT ANSWER 20. Which one of the following must be true about the program? (A) George performs X. (B) Helen performs O. (C) Helen performs T. (D) Leslie performs P. (E) Leslie performs X. (A) Violates Inf 3 (B) Violates Rule 4 (C) CORRECT ANSWER (D) It is a can not a must (L: OP ok but so is L: OX) (E) It is a can not a must (L: OP ok but so is L: OX) 21. Which one of the following is a complete and accurate list of the songs any one of which could be the last song performed at the concert? (A) O, P, Z (B) O, T, X (C) T, P, Z (D) T, X, Z (E) X, P, Z (A) Violates Rule 2 (O and P can t occur after Z so they can t be the last song) (B) Violates Rule 2 (O can t occur after Z so they can t be the last song) (C) Violates Rule 2 (P can t occur after Z so they can t be the last song) (D) CORRECT ANSWER (E) Violates Rule 2 (P can t occur after Z so they can t be the last song) 22. If X is performed first, which one of the following must be true? (A) X is performed by George. (B) X is performed by Helen. (C) P is the fourth song performed. (D) Y is the second song performed. (E) Y is the third song performed. (D) CORRECT ANSWER. Because of Inf 1, there are 4 songs that have to come after Y (O, P, T, and Z). If X is first, then Y must be 2nd so the 4 other songs can happen after Y. 23. Each of the following is an acceptable schedule for the performance of the songs, in order from the first to last song performed, EXCEPT: (A) Y, O, P, T, Z, X (B) Y, T, O, P, X, Z (C) Y, X, O, P, Z, T (D) X, Y, O, P, Z, T (E) X, Y, O, T, P, Z (B) CORRECT ANSWER. Z can t be last if Y is first because Inf 3.

8 24. If Y is performed first, the songs performed second, third, and fourth, respectively, could be (A) T, X, and O (B) T, Z, and O (C) X, O, and P (D) X, P, and Z (E) X, T, and O Y??? (A) Violates Inf 3 Y T X O P Z (B) Violates Rule 2 Y T Z O (C) CORRECT ANSWER Y X O P Z T (D) Violates Rule 2 Y X P Z O or O (E) Violates Inf 3 Y X T O P Z QA4 Answers: Questions A 2. A 3. D 4. A 5. B Questions A 7. E 8. B 9. A 10. A 11. C 12. E Questions C 14. D 15. C 16. E 17. E 18. D Questions E 20. C 21. D 22. D 23. B 24. C

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