Question: 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?

Answer: O, P, Z

Answer: O, T, X

Answer: T, P, Z

Answer: T, X, Z

Answer: X, P, Z


Context: 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.

Question: If X is performed first, which one of the following must be true?

Answer: X is performed by George.

Answer: X is performed by Helen.

Answer: P is the fourth song performed.

Answer: Y is the second song performed.

Answer: Y is the third song performed.


Question: Each of the following is an acceptable schedule for the performance of the songs, in order from the first to last song performed, EXCEPT:

Answer: Y, O, P, T, Z, X

Answer: Y, T, O, P, X, Z

Answer: Y, X, O, P, Z, T

Answer: X, Y, O, P, Z, T

Answer: X, Y, O, T, P, Z


Question: If Y is performed first, the songs performed second, third, and fourth, respectively, could be

Answer: T, X, and O

Answer: T, Z, and O

Answer: X, O, and P

Answer: X, P, and Z

Answer: X, T, and O


Context: Seven students—fourth-year students Kim and Lee; third-year students Pat and Robin; and second-year students Sandy, Terry, and Val—and only those seven, are being assigned to rooms of equal size in a dormitory. Each room assigned must have either one, or two, or three students assigned to it, and will accordingly be called either a single, or a double, or a triple. The seven students are assigned to rooms in accordance with the following conditions: No fourth-year student can be assigned to a triple. No second-year student can be assigned to a single. Lee and Robin must not share the same room. Kim and Pat must share the same room.

Question: Which one of the following is a combination of rooms to which the seven students could be assigned?

Answer: two triples and one single

Answer: one triple and four singles

Answer: three doubles and a single

Answer: two doubles and three singles

Answer: one double and five singles


Question: If the room assigned to Robin is a single, which one of the following could be true?

Answer: There is exactly one double that has a second-year student assigned to it.

Answer: Lee is assigned to a single.

Answer: Sandy, Pat and one other student are assigned to a triple together.

Answer: Exactly three of the rooms assigned to the students are singles.

Answer: Exactly two of the rooms assigned to the students are doubles.


Question: Which one of the following must be true?

Answer: Lee is assigned to a single.

Answer: Pat shares a double with another student.

Answer: Robin shares a double with another student.

Answer: Two of the second-year students share a double with each other.

Answer: Neither of the third-year students is assigned to a single.


Question: If Robin is assigned to a triple, which one of the following must be true?

Answer: Lee is assigned to a single.

Answer: Two second-year students share a double with each other.

Answer: None of the rooms assigned to the students is a single.

Answer: Two of the rooms assigned to the students are singles.

Answer: Three of the rooms assigned to the students are singles.


Question: If Terry and Val are assigned to different doubles from each other, then it must be true of the students' rooms that exactly

Answer: one is a single

Answer: two are singles

Answer: two are doubles

Answer: one is a triple

Answer: two are triples


Question: Which one of the following could be true?

Answer: The two fourth-year students are assigned to singles.

Answer: The two fourth-year students share a double with each other.

Answer: Lee shares a room with a second-year student.

Answer: Lee shares a room with a third-year student

Answer: Pat shares a triple with two other students.


Context: A worker will insert colored light bulbs into a billboard equipped with exactly three light sockets, which are labeled lights 1, 2, and 3. The worker has three green bulbs, three purple bulbs, and three yellow bulbs. Selection of bulbs for the sockets is governed by the following conditions: Whenever light 1 is purple, light 2 must be yellow. Whenever light 2 is green, light 1 must be green. Whenever light 3 is either purple or yellow, light 2 must be purple.

Question: Which one of the following could be an accurate list of the colors of light bulbs selected for lights 1, 2, and 3, respectively?

Answer: green, green, yellow

Answer: purple, green, green

Answer: purple, purple, green

Answer: yellow, purple, green

Answer: yellow, yellow, yellow


Question: If light 1 is yellow, then any of the following can be true, EXCEPT:

Answer: Light 2 is green.

Answer: Light 2 is purple.

Answer: Light 3 is green.

Answer: Light 3 is purple.

Answer: Light 3 is yellow.


Question: There is exactly one possible color sequence of the three lights if which one of the following is true?

Answer: Light 1 is purple.

Answer: Light 2 is purple.

Answer: Light 2 is yellow.

Answer: Light 3 is purple.

Answer: Light 3 is yellow.


Question: If no green bulbs are selected, there are exactly how many possible different color sequences of the three lights?

Answer: one

Answer: two

Answer: three

Answer: four

Answer: five


Question: If no two lights are assigned light bulbs that are the same color as each other, then which one of the following could be true?

Answer: Light 1 is green, and light 2 is purple.

Answer: Light 1 is green, and light 2 is yellow.

Answer: Light 1 is purple, and light 3 is yellow.

Answer: Light 1 is yellow, and light 2 is green.

Answer: Light 1 is yellow, and light 3 is purple.


Context: An attorney is scheduling interviews with witnesses for a given week, Monday through Saturday. Two full consecutive days of the week must be reserved for interviewing hostile witnesses. In addition, nonhostile witnesses Q, R, U, X, Y, and Z will each be interviewed exactly once for a full morning or afternoon. The only witnesses who will be interviewed simultaneously with each other are Q and R. The following conditions apply: X must be interviewed on Thursday morning. Q must be interviewed at some time before X. U must be interviewed at some time before R. Z must be interviewed at some time after X and at some time after Y.

Question: Which one of the following is a sequence, from first to last, in which the nonhostile witnesses could be interviewed?

Answer: Q with R, U, X, Y, Z

Answer: Q, U, R, X with Y, Z

Answer: U, X, Q, with R, X, Z

Answer: U, Y, Q with R, X, Z

Answer: X, Q with U, Z, R, Y


Question: Which one of the following is acceptable as a complete schedule of witnesses for Tuesday morning, Tuesday afternoon, and Wednesday morning, respectively?

Answer: Q, R, none

Answer: R, none, Y

Answer: U, none, X

Answer: U, Y, none

Answer: Y, Z, none


Question: If Y is interviewed at some time after X, which one of the following must be a day reserved for interviewing hostile witnesses?

Answer: Monday

Answer: Tuesday

Answer: Wednesday

Answer: Friday

Answer: Saturday


Question: If R is interviewed at some time after Y, which one of the following must be a day reserved for interviewing hostile witnesses?

Answer: Monday

Answer: Tuesday

Answer: Wednesday

Answer: Thursday

Answer: Friday


Question: If on Wednesday afternoon and on Monday the attorney conducts no interviews, which one of the following must be true?

Answer: Q is interviewed on the same day as U.

Answer: R is interviewed on the same day as Y.

Answer: Y is interviewed at some time before U.

Answer: Y is interviewed at some time before Wednesday.

Answer: Z is interviewed at some time before Friday.


Question: If Z is interviewed on Saturday morning, which one of the following can be true?

Answer: Wednesday is a day reserved for interviewing hostile witnesses.

Answer: Friday is a day reserved for interviewing hostile witnesses.

Answer: R is interviewed on Thursday.

Answer: U is interviewed on Tuesday.

Answer: Y is interviewed at some time before Thursday.


Context: During a four-week period, each of seven previously unadvertised products—G, H, J, K, L, M, and O—will be advertised. A different pair of these products will be advertised each week. Exactly one of the products will be a member of two of these four pairs. The following constraints must be observed: J is not advertised during a given week unless H is advertised during the immediately preceding week. The product that is advertised during two of the weeks is advertised during week 4 but is not advertised during week 3. G is not advertised during a given week unless either J or else O is also advertised that week. K is advertised during one of the first two weeks. O is one of the products advertised during week 3.

Question: Which one of the following could be the schedule of advertisements?

Answer: week 1: G, J; week 2: K, L; week 3: O, M; week 4: H, L

Answer: week l: H, K; week 2: J, G; week 3: O, L; week 4: M, K

Answer: week 1: H, K; week 2: J, M; week 3: O, L; week 4: G, M

Answer: week 1: H, L; week 2: J, M; week 3: O, G; week 4: K, L

Answer: week 1: K, M; week 2: H, J; week 3: O, G; week 4: L, M


Question: Which one of the following is a pair of products that CANNOT be advertised during the same week as each other?

Answer: H and K

Answer: H and M

Answer: J and O

Answer: K and L

Answer: L and M


Question: Which one of the following must be advertised during week 2?

Answer: G

Answer: J

Answer: K

Answer: L

Answer: M


Question: Which one of the following CANNOT be the product that is advertised during two of the weeks?

Answer: G

Answer: H

Answer: K

Answer: L

Answer: M


Question: If L is the product that is advertised during two of the weeks, which one of the following is a product that must be advertised during one of the weeks in which L is advertised?

Answer: G

Answer: H

Answer: J

Answer: K

Answer: M


Question: Which one of the following is a product that could be advertised in any of the four weeks?

Answer: H

Answer: J

Answer: K

Answer: L

Answer: O


Question: Which one of the following is a pair of products that could be advertised during the same week as each other?

Answer: G and H

Answer: H and J

Answer: H and O

Answer: K and O

Answer: M and O


Context: An upholsterer will do seven jobs—N, O, P, S, T, U, and X—during a given week on Monday through Friday. Each job is done on exactly one of the days, and no two jobs are done concurrently. Each job takes either one half day or an entire day. The following conditions apply: N is done on the same day as S. X is done on Wednesday, and T is done on Friday. S is done earlier than the day on which P is done. U and X are the only jobs that must take an entire day each.

Question: Which one of the following is an acceptable schedule for the seven jobs? Mon. Tues. Wed. Thurs. Fri.

Answer: N, S P X O, U T

Answer: O N, S X P U

Answer: P U X N, S O, T

Answer: S N, P X U O, T

Answer: U N, S X O P, T


Question: Which one of the following is a complete and accurate list of those jobs each of which CANNOT be done on Tuesday?

Answer: P, U

Answer: T, X

Answer: N, P, S

Answer: N, S, X

Answer: T, U, X


Question: If O is done on Thursday, N could be done on

Answer: Monday, but on no other day

Answer: Thursday, but on no other day

Answer: Monday or else on Tuesday, but on no other day

Answer: Tuesday or else on Wednesday, but on no other day

Answer: Wednesday or else on Thursday, but on no other day


Question: Which one of the following must be true if the upholsterer does two jobs on Thursday and two jobs on Friday?

Answer: N is done on Monday.

Answer: O is done on Tuesday.

Answer: P is done on Friday.

Answer: O is done on the same day as P.

Answer: O is done on the same day as T.


Question: If P is not done on Friday, which one of the following must be true?

Answer: N is done on Monday or else on Thursday.

Answer: O is done on Wednesday or else on Thursday.

Answer: S is done on Monday or else on Thursday.

Answer: P is done on the same day as O.

Answer: O is done on the same day as either P or T.


Question: If O is done on Monday, which one of the following must be true?

Answer: N is done on Tuesday.

Answer: P is done on Thursday.

Answer: P is done on Friday.

Answer: S is done on Thursday.

Answer: U is done on Friday.


Context: Seven persons—G, H, I, K, L, M, and P—are going rock-climbing together. One person will be the organizer and not a member of any team. The remaining six will form three climbing teams, each consisting of two persons. No person can be on more than one team. The following conditions also apply: M and P cannot be on the same team as each other. If K is on a team, M must also be on that team. If G is on a team, either H or I must also be on that team. If H is on a team, K must be the organizer.

Question: Which one of the following can be a list of the members of the three teams?

Answer: Team 1: G, H; Team 2: I, L; Team 3: M, P

Answer: Team 1: G, H; Team 2: L, P; Team 3: K, M

Answer: Team 1: G, I; Team 2: H, M; Team 3: L, P

Answer: Team 1: G, P; Team 2: I, L; Team 3: K, M

Answer: Team 1: H, M; Team 2: I, L; Team 3: K, P


Question: Each of the following is a pair of persons who can constitute a team EXCEPT:

Answer: H, I

Answer: H, M

Answer: I, M

Answer: I, P

Answer: L, P


Question: Which one of the following can be the organizer, who is not on any team?

Answer: G

Answer: H

Answer: I

Answer: L

Answer: P


Question: If G and I form a team and if H is a member of a team, then H must be on the same team as either

Answer: K or L

Answer: K or P

Answer: L or M

Answer: L or P

Answer: M or P


Question: If L and M are on different teams, which one of the following must be true?

Answer: H is the organizer.

Answer: K is the organizer.

Answer: G is on the same team as H.

Answer: I is on the same team as M.

Answer: L is on the same team as P.


Question: Which one of the following lists three persons each of whom must be on a team and no two of whom can be on the same team as each other?

Answer: G, L, M

Answer: G, M, P

Answer: H, I, L

Answer: I, L, P

Answer: L, M, P


Question: There are how many different persons any one of whom could be paired with M to form a team?

Answer: one

Answer: two

Answer: three

Answer: four

Answer: five


Context: A train makes five trips around a loop through five stations—P, Q, R, S, and T, in that order—stopping at exactly three of the stations on each trip. The train must conform to the following conditions: The train stops at any given station on exactly three trips, but not on three consecutive trips. The train stops at any given station at least once in any two consecutive trips.

Question: Which one of the following could be the list of stations at which the train stops on the first two trips?

Answer: first trip: P, Q, S; second trip: P, Q, R

Answer: first trip: P, Q, T; second trip: Q, R, T

Answer: first trip: Q, R, S; second trip: P, Q, S

Answer: first trip: Q, S, T; second trip: P, R, S

Answer: first trip: R, S, T; second trip: P, R, T


Question: If on the first and third trips the train is to stop at Q, R, and S, and at R, S, and T, respectively, which one of the following is the list of stations at which it must stop on the second trip?

Answer: P, Q, R

Answer: P, Q, T

Answer: P, S, T

Answer: Q, R, T

Answer: Q, S, T


Question: The train CANNOT stop at both P and Q on both the

Answer: first and third trips

Answer: first and fourth trips

Answer: second and fourth trips

Answer: second and fifth trips

Answer: fourth and fifth trips


Question: If the train is to stop at Q, R, and T on the first trip and at Q, R, and S on the fourth trip, then it could be true that the train

Answer: stops at Q on the second trip

Answer: stops at R on the fifth trip

Answer: stops at T on the second trip

Answer: does not stop at P on the third trip

Answer: does not stop at T on the fifth trip


Question: Suppose that the train's scheduler mistakenly scheduled the train to make the following stops on the first three trips: Q, R, and S on the first trip; P, Q, and S on the second trip; and R, S, and T on the third trip. Which one of the following is a substitution that results in a schedule for those three trips under which the train violates no condition?

Answer: on the first trip, P for Q

Answer: on the second trip, R for S

Answer: on the second trip, T for S

Answer: on the third trip, P for R

Answer: on the third trip, P for S


Context: On a certain morning, each of six speakers—Feinberg, Guzman, Harrison, Jansen, Kim, and Mackey—will lecture for exactly one hour. Each lecture will take place in either the library or the studio, with exactly one speaker at a time lecturing in each of these two rooms. The lectures must be given in a manner consistent with the following conditions: Exactly three of the lectures are given in each room, the first beginning at precisely 8 A.M. , the second at precisely 9 A.M., and the third at precisely 10 A.M. Feinberg's lecture cannot begin earlier than Guzman's. Neither Jansen's lecture nor Mackey's lecture begins at the same time as Feinberg's. Harrison's lecture begins earlier than Feinberg's. Jansen and Mackey, not necessarily in that order, lecture in the studio.

Question: Which one of the following must be true?

Answer: Either Jansen or else Mackey lectures in the studio beginning at 8 A.M.

Answer: Either Harrison or else Mackey lectures in the studio beginning at 9 A.M.

Answer: Either Guzman or else Harrison lectures in the library beginning at 8 A.M.

Answer: Either Feinberg or else Guzman lectures in the library beginning at 9 A.M.

Answer: Either Feinberg or else Guzman lectures in the library beginning at 10 A.M.


Question: If both Kim's lecture and Mackey's lecture begin at 9 A.M. , which one of the following could be true?

Answer: Guzman's lecture begins at 8 A.M.

Answer: Harrison's lecture begins at 10 A.M.

Answer: Jansen's lecture begins at 10 A.M.

Answer: Guzman lectures in the library.

Answer: Harrison lectures in the studio.


Question: If Harrison's lecture begins at 9 A.M. , which one of the following must be true?

Answer: Feinberg lectures in the library.

Answer: Guzman lectures in the studio.

Answer: Harrison lectures in the library.

Answer: Kim lectures in the studio.

Answer: Mackey lectures in the library.


Question: Each of the following is a pair of speakers whose lectures could begin at the same time as each other EXCEPT:

Answer: Feinberg and Guzman

Answer: Feinberg and Kim

Answer: Guzman and Jansen

Answer: Guzman and Kim

Answer: Guzman and Mackey


Question: If Feinberg lectures in the library beginning at 9 A.M., which one of the following could be true?

Answer: Guzman lectures in the studio beginning at 10 A.M.

Answer: Harrison lectures in the studio beginning at 8 A.M.

Answer: Jansen lectures in the studio beginning at 10 A.M.

Answer: Kim lectures in the studio beginning at 9 A.M.

Answer: Mackey lectures in the studio beginning at 9 A.M.


Question: Which one of the following could be a list of the speakers who give their lectures in the studio, in the order in which they give their lectures, from first to third, respectively?

Answer: Feinberg, Jansen, Mackey

Answer: Guzman, Kim, Mackey

Answer: Harrison, Jansen, Kim

Answer: Jansen, Mackey, Harrison

Answer: Mackey, Feinberg, Jansen


Context: At a benefit dinner, a community theater's seven sponsors—K, L, M, P, Q, V, and Z—will be seated at three tables—1, 2, and 3. Of the sponsors, only K, L, and M will receive honors, and only M, P, and Q will give a speech. The sponsors' seating assignments must conform to the following conditions: Each table has at least two sponsors seated at it, and each sponsor is seated at exactly one table. Any sponsor receiving honors is seated at table 1 or table 2. L is seated at the same table as V.

Question: Which one of the following is an acceptable assignment of sponsors to tables?

Answer: Table 1: K, P; Table 2: M, Q; Table 3: L, V, Z

Answer: Table 1: K, Q, Z; Table 2: L, V; Table 3: M, P

Answer: Table 1: L, P; Table 2: K, M; Table 3: Q, V, Z

Answer: Table 1: L, Q, V; Table 2: K, M; Table 3: P, Z

Answer: Table 1: L, V, Z; Table 2: K, M, P; Table 3: Q


Question: Which one of the following is a list of all and only those sponsors any one of whom could be among the sponsors assigned to table 3?

Answer: P, Q

Answer: Q, Z

Answer: P, Q, Z

Answer: Q, V, Z

Answer: P, Q, V, Z


Question: If K is assigned to a different table than M, which one of the following must be true of the seating assignment?

Answer: K is seated at the same table as L.

Answer: L is seated at the same table as Q.

Answer: M is seated at the same table as V.

Answer: Exactly two sponsors are seated at table 1.

Answer: Exactly two sponsors are seated at table 3.


Question: If Q is assigned to table 1 along with two other sponsors, which one of the following could be true of the seating assignment?

Answer: K is seated at the same table as L.

Answer: K is seated at the same table as Q.

Answer: M is seated at the same table as V.

Answer: M is seated at the same table as Z.

Answer: P Is seated at the same table as Q.


Question: If the sponsors assigned to table 3 include exactly one of the sponsors who will give a speech, then the sponsors assigned to table 1 could include any of the following EXCEPT:

Answer: K

Answer: M

Answer: P

Answer: Q

Answer: Z


Question: If three sponsors, exactly two of whom are receiving honors, are assigned to table 2, which one of the following could be the list of sponsors assigned to table 1?

Answer: K, M

Answer: K, Z

Answer: P, V

Answer: P, Z

Answer: Q, Z


Question: Which one of the following conditions, if added to the existing conditions, results in a set of conditions to which no seating assignment for the sponsors can conform?

Answer: At most two sponsors are seated at table 1.

Answer: Any sponsor giving a speech is seated at table 1 or else table 2.

Answer: Any sponsor giving a speech is seated at table 2 or else table 3.

Answer: Exactly three of the sponsors are seated at table 1.

Answer: Any table at which both L and V are seated also has a third sponsor seated at it.


Context: Exactly four medical training sessions—M, O, R, and S—will be scheduled for four consecutive days—day 1 through day 4—one session each day. Six professionals—three nurses and three psychologists—will teach the sessions. The nurses are Fine, Johnson, and Leopold; the psychologists are Tyler, Vitale, and Wong. Each session will be taught by exactly one nurse and exactly one psychologist. The schedule must conform to the following conditions: Each professional teaches at least once. Day 3 is a day on which Leopold teaches. Neither Fine nor Leopold teaches with Tyler. Johnson teaches session S only. Session M is taught on the day after the day on which session S is taught.

Question: If session R is the only session for which Leopold is scheduled, which one of the following is a pair of professionals who could be scheduled for day 2 together?

Answer: Fine and Tyler

Answer: Fine and Wong

Answer: Johnson and Tyler

Answer: Johnson and Wong

Answer: Leopold and Wong


Question: Which one of the following must be false?

Answer: Session O is scheduled for day 1.

Answer: Session S is scheduled for day 3.

Answer: Leopold is scheduled for day 1.

Answer: Vitale is scheduled for day 4.

Answer: Wong is scheduled for day 1.


Question: Which one of the following could be the session and the professionals scheduled for day 4 ?

Answer: session M, Fine, Wong

Answer: session O, Fine, Tyler

Answer: session O, Johnson, Tyler

Answer: session R, Fine, Wong

Answer: session S, Fine, Vitale


Question: If session S is scheduled for day 2, which one of the following is a professional who must be scheduled to teach session M ?

Answer: Fine

Answer: Leopold

Answer: Tyler

Answer: Vitale

Answer: Wong


Question: If session O and session R are scheduled for consecutive days, which one of the following is a pair of professionals who could be scheduled for day 2 together?

Answer: Fine and Leopold

Answer: Fine and Wong

Answer: Johnson and Tyler

Answer: Johnson and Vitale

Answer: Leopold and Tyler


Question: Which one of the following could be the order in which the nurses teach the sessions, listed from day 1 through day 4?

Answer: Fine, Johnson, Leopold, Leopold

Answer: Fine, Leopold, Leopold, Johnson

Answer: Johnson, Johnson, Leopold, Fine

Answer: Johnson, Leopold, Leopold, Johnson

Answer: Leopold, Leopold, Fine, Fine


Question: If session O is scheduled for day 3, which one of the following must be scheduled for day 4?

Answer: session R

Answer: session S

Answer: Fine

Answer: Leopold

Answer: Vitale


Context: Six paintings hang next to each other as shown below: Each of the paintings is an oil or else a watercolor. Each oil is directly beside, directly above, or directly below another oil. Each watercolor is directly beside, directly above, or directly below another watercolor. Each painting is a nineteenth-century painting or else a twentieth-century painting. Each painting is directly beside, directly above, or directly below another painting painted in the same century. Painting 2 is a nineteenth-century painting. Painting 3 is an oil. Painting 5 is a twentieth-century painting.

Question: If all of the nineteenth-century paintings are watercolors, which one of the following must be true?

Answer: Painting 1 is an oil.

Answer: Painting 3 is a nineteenth-century painting.

Answer: Painting 4 is a watercolor.

Answer: Painting 5 is an oil.

Answer: Painting 6 is a twentieth-century painting.


Question: It is possible that the only two watercolors among the six paintings are

Answer: paintings 1 and 5

Answer: paintings 1 and 6

Answer: paintings 2 and 4

Answer: paintings 4 and 5

Answer: paintings 4 and 6


Question: If there are exactly three oils and three watercolors, which one of the following must be true?

Answer: Painting 1 is a watercolor.

Answer: Painting 2 is a watercolor.

Answer: Painting 4 is a watercolor.

Answer: Painting 5 is a watercolor.

Answer: Painting 6 is a watercolor.


Question: If exactly two paintings are oils and exactly two paintings are nineteenth-century paintings, which one of the following must be false?

Answer: Painting 1 is a nineteenth-century painting, and painting 6 is an oil.

Answer: Painting 2 is both a nineteenth-century painting and an oil.

Answer: Painting 3 is a nineteenth-century painting.

Answer: Paintings 1 and 2 are both nineteenth-century paintings.

Answer: Painting 2 is an oil, and painting 4 is a nineteenth-century painting.


Question: Which one of the following could be true?

Answer: Paintings 1 and 4 are two of exactly three twentieth-century paintings.

Answer: Paintings 1 and 6 are two of exactly three twentieth-century paintings.

Answer: Paintings 1 and 6 are two of exactly three nineteenth-century paintings.

Answer: Paintings 3 and 4 are two of exactly three nineteenth-century paintings.

Answer: Paintings 4 and 6 are two of exactly three nineteenth-century paintings.


Context: Exactly six of seven jugglers—G, H, K, L, N, P, and Q—are each assigned to exactly one of three positions—front, middle, and rear—on one of two teams—team 1 and team 2. For each team, exactly one juggler must be assigned to each position according to the following conditions: If either G or H or both are assigned to teams, they are assigned to front positions. If assigned to a team, K is assigned to a middle position. If assigned to a team, L is assigned to team 1. Neither P nor K is on the same team as N. P is not on the same team as Q. If H is on team 2, then Q is assigned to the middle position on team 1.

Question: Which one of the following is an acceptable list of assignments of jugglers to team 2?

Answer: front: Q; middle: K; rear: N

Answer: front: H; middle: P; rear: K

Answer: front: H; middle: L; near: N

Answer: front: G; middle: Q; rear: P

Answer: front: G; middle: Q; rear: N


Question: If H is assigned to team 2, which one of the following is an acceptable assignment of jugglers to team 1?

Answer: front: G; middle: K; rear: L

Answer: front: G; middle: K; rear: N

Answer: front: L; middle: K; rear: P

Answer: front: L; middle: Q; rear: G

Answer: front: L; middle: Q; rear: N


Question: Which one of the following is an acceptable list of assignments of jugglers to team 1?

Answer: front: G; middle: K; rear: L

Answer: front: G; middle: K; rear: P

Answer: front: L; middle: K; rear: Q

Answer: front: Q; middle: K; rear: P

Answer: front: Q; middle: L; rear: N


Question: If G is assigned to team 1, which one of the following is a pair of jugglers who could also be assigned to team 1?

Answer: H and N

Answer: K and L

Answer: K and P

Answer: L and N

Answer: L and Q


Question: If G is assigned to team 1 and K is assigned to team 2, which one of the following must be assigned the rear position on team 2?

Answer: H

Answer: L

Answer: N

Answer: P

Answer: Q


Context: A producer is positioning exactly seven music pieces—F, G, H, J, K, L, and M—one after another on a music recording, not necessarily in that order. Each piece will fill exactly one of the seven sequential tracks on the recording, according to the following conditions: F must be second. J cannot be seventh. G can come neither immediately before nor immediately after H. H must be in some track before that of L. L must be in some track before that of M.

Question: Which one of the following could be the order, from first to seventh, of the pieces on the recording?

Answer: F, K, G, J, H, L, M

Answer: G, F, H, K, L, J, M

Answer: G, F, H, K, L, M, J

Answer: K, F, G, H, J, L, M

Answer: K, F, L, J, H, M, G


Question: If M fills some track before that of J and also before that of K on the recording, which one of the following must be true?

Answer: G is first.

Answer: K is seventh.

Answer: L is third.

Answer: H comes either immediately before or immediately after F.

Answer: L comes either immediately before or immediately after G.


Question: Which one of the following is a complete and accurate list of the pieces any of which could be first on the recording?

Answer: G, J, K

Answer: G, H, J, K

Answer: G, H, J, L

Answer: G, J, K, L

Answer: H, J, K, L, M


Question: The earliest track that M can fill is the

Answer: first

Answer: third

Answer: fourth

Answer: fifth

Answer: sixth


Question: If G is to come immediately before H but all the other conditions remain in effect, any of the following could be true EXCEPT:

Answer: J comes immediately before F.

Answer: K comes immediately before G.

Answer: J comes immediately after L.

Answer: J comes immediately after K.

Answer: K comes immediately after M.


Context: Fu, Gunsel, Jackson, Kowalski, Lee, Mayer, and Ordoveza are the only applicants being considered for some positions at a nonprofit organization. Only applicants who are interviewed will be hired. The hiring process must meet the following constraints: If Gunsel is interviewed, Jackson is interviewed. If Jackson is interviewed, Lee is interviewed. Fu is interviewed. Fu is not hired, unless Kowalski is interviewed. Kowalski is not hired, unless Mayer is interviewed. If Mayer is hired, and Lee is interviewed, Ordoveza is hired.

Question: Which one of the following could be a complete and accurate list of the applicants that are interviewed?

Answer: Fu, Gunsel

Answer: Fu, Jackson

Answer: Fu, Lee

Answer: Fu, Gunsel, Lee

Answer: Fu, Gunsel, Jackson


Question: Which one of the following could be true?

Answer: Lee and Mayer are the only applicants interviewed.

Answer: Fu, Jackson, and Kowalski are the only applicants interviewed.

Answer: Gunsel and one other applicant are the only applicants interviewed.

Answer: Gunsel and two other applicants are the only applicants interviewed.

Answer: Gunsel and three other applicants are the only applicants interviewed.


Question: If Mayer is not interviewed, which one of the following must be true?

Answer: Kowalski is not interviewed.

Answer: Kowalski is interviewed but not hired.

Answer: Fu is not hired.

Answer: Fu is hired but Kowalski is not hired.

Answer: Fu is interviewed but Kowalski is not hired.


Question: If Gunsel and five other applicants are the only applicants interviewed, and if exactly three applicants are hired, then which one of the following could be an accurate list of the applicants hired?

Answer: Fu, Lee, Mayer

Answer: Fu, Kowalski, Mayer

Answer: Kowalski, Lee, Ordoveza

Answer: Gunsel, Jackson, Mayer

Answer: Gunsel, Jackson, Lee


Question: If every applicant that is interviewed is hired, and if Lee is hired, then each of the following applicants must be interviewed EXCEPT:

Answer: Fu

Answer: Jackson

Answer: Kowalski

Answer: Mayer

Answer: Ordoveza


Question: If Ordoveza is not interviewed, and if exactly four applicants are hired, then which one of the following must be false?

Answer: Lee is hired.

Answer: Mayer is hired.

Answer: Jackson is interviewed.

Answer: Kowalski is interviewed.

Answer: Gunsel is interviewed.


Context: Exactly six of seven researchers—three anthropologists: Franklin, Jones, and Marquez; and four linguists: Neil, Osborne, Rice, and Samuels—will be included in two three-person teams—team 1 and team 2. No researcher will be included in more than one team. Each team must include at least one anthropologist and at least one linguist. The teams' composition must conform to the following conditions: Neither team includes both Franklin and Samuels. Neither team includes both Neil and Rice. If a team includes Marquez, it includes neither Rice nor Samuels. If team 1 includes Jones, team 2 includes Rice.

Question: Which one of the following could be the list of the researchers on the two teams?

Answer: team 1: Franklin, Marquez, Osborne team 2: Jones, Neil, Rice

Answer: team 1: Franklin, Neil, Samuels team 2: Jones, Osborne, Rice

Answer: team 1: Franklin, Osborne, Rice team 2: Jones, Neil, Samuels

Answer: team 1: Jones, Marquez, Neil team 2: Osborne, Rice, Samuels

Answer: team 1: Jones, Osborne, Rice team 2: Franklin, Marquez, Neil


Question: If Jones is on team 1, which one of the following is a pair of researchers that must be on team 2 together?

Answer: Franklin and Rice

Answer: Marquez and Osborne

Answer: Neil and Osborne

Answer: Osborne and Samuels

Answer: Rice and Samuels


Question: If Neil is on team 1, which one of the following is a pair of researchers that could be on team 1 together with Neil?

Answer: Franklin and Jones

Answer: Jones and Osborne

Answer: Jones and Rice

Answer: Jones and Samuels

Answer: Osborne and Samuels


Question: If Franklin is on the same team as Marquez, which one of the following could be true?

Answer: Jones is on team 1.

Answer: Rice is on team 1.

Answer: Samuels is on team 2.

Answer: Both Neil and Osborne are on team 1.

Answer: Both Neil and Osborne are on team 2.


Question: Each of the following is a pair of researchers that could be on team 2 together EXCEPT:

Answer: Franklin and Jones

Answer: Franklin and Marquez

Answer: Franklin and Rice

Answer: Jones and Marquez

Answer: Jones and Rice


Question: Which one of the following could be true?

Answer: Franklin is on team 1 and Neil is on team 2.

Answer: Franklin is on team 2 and Jones is not on any team.

Answer: Franklin is on team 2 and Marquez is on team 2.

Answer: Franklin is not on any team and Jones is on team 1.

Answer: Jones is on team 1 and Neil is on team 2.


Question: If Marquez is on team 2, which one of the following must also be on team 2?

Answer: Franklin

Answer: Jones

Answer: Osborne

Answer: Rice

Answer: Samuels


Context: Five candidates for mayor—Q, R, S, T, and U—will each speak exactly once at each of three town meetings—meetings 1, 2, and 3. At each meeting, each candidate will speak in one of five consecutive time slots. No two candidates will speak in the same time slot as each other at any meeting. The order in which the candidates will speak will meet the following conditions: Each candidate must speak either first or second at at least one of the meetings. Any candidate who speaks fifth at any of the meetings must speak first at at least one of the other meetings. No candidate can speak fourth at more than one of the meetings.

Question: Which one of the following could be the order, from first to fifth, in which the candidates speak at the meetings?

Answer: meeting 1: Q, U, R, T, S meeting 2: S, T, R, U, Q meeting 3: T, U, Q, R, S

Answer: meeting 1: R, S, Q, T, U meeting 2: U, T, S, R, Q meeting 3: Q, R, T, U, S

Answer: meeting 1: S, Q, U, T, R meeting 2: U, T, Q, R, S meeting 3: R, Q, S, T, U

Answer: meeting 1: T, R, S, U, Q meeting 2: Q, R, S, T, U meeting 3: U, S, R, Q, T

Answer: meeting 1: U, T, R, S, Q meeting 2: Q, R, S, T, U meeting 3: S, T, U, Q, R


Question: If R speaks second at meeting 2 and first at meeting 3, which one of the following is a complete and accurate list of those time slots any one of which could be the time slot in which R speaks at meeting 1?

Answer: fourth, fifth

Answer: first, second, fifth

Answer: second, third, fifth

Answer: third, fourth, fifth

Answer: second, third, fourth, fifth


Question: If the order in which the candidates speak at meeting 1 is R, U, S, T, Q, and the order in which they speak at meeting 2 is Q, R, U, S, T, which one of the following could be true of meeting 3?

Answer: Q speaks first.

Answer: R speaks third.

Answer: S speaks first.

Answer: T speaks second.

Answer: U speaks fifth.


Question: If R speaks first at meetings 1 and 2, and S speaks first at meeting 3, which one of the following must be true?

Answer: R speaks second at meeting 3.

Answer: R speaks fourth at meeting 3.

Answer: S speaks second at at least one of the meetings.

Answer: S speaks fifth at exactly one of the meetings.

Answer: S speaks fifth at exactly two of the meetings.


Question: It could be true that at all three meetings T speaks

Answer: first

Answer: second

Answer: in some time slot after the time slot in which R speaks

Answer: in some time slot after the time slots in which S and U speak

Answer: in some time slot before the time slots in which R and U speak


Question: If S, T, and U speak second at meetings 1, 2, and 3, respectively, which one of the following must be true?

Answer: The fifth speaker at at least one of the meetings is either Q or R.

Answer: Either Q speaks first at exactly two of the meetings or else R does so.

Answer: Neither S nor T speaks fifth at any of the meetings.

Answer: Q speaks third at one of the meetings, and R speaks third at another of the meetings.

Answer: Q speaks fourth at one of the meetings, and R speaks fourth at another of the meetings.


Context: In a certain recipe contest, each contestant submits two recipes, one for an appetizer and one for a main dish. Together the two recipes must include exactly seven flavorings—fenugreek, ginger, lemongrass, nutmeg, paprika, saffron, and turmeric—with no flavoring included in more than one of the two recipes. Each contestant's recipes must satisfy the following conditions: The appetizer recipe includes at most three of the flavorings. Fenugreek is not included in the same recipe as nutmeg. Saffron is not included in the same recipe as turmeric. Ginger is included in the same recipe as nutmeg.

Question: Which one of the following could be a complete and accurate list of the flavorings included in one contestant's main-dish recipe?

Answer: fenugreek, lemongrass, saffron

Answer: fenugreek, ginger, nutmeg, turmeric

Answer: ginger, lemongrass, nutmeg, paprika

Answer: ginger, nutmeg, paprika, turmeric

Answer: lemongrass, nutmeg, saffron, turmeric


Question: If a contestant's appetizer recipe does not include fenugreek, then the contestant's appetizer recipe must include

Answer: ginger

Answer: lemongrass

Answer: paprika

Answer: saffron

Answer: turmeric


Question: Which one of the following could be a list of all of the flavorings included in one contestant's appetizer recipe?

Answer: fenugreek, saffron

Answer: ginger, nutmeg

Answer: fenugreek, nutmeg, turmeric

Answer: lemongrass, nutmeg, saffron

Answer: fenugreek, lemongrass, paprika, turmeric


Question: If a contestant includes lemongrass in the same recipe as paprika, which one of the following is a flavoring that must be included in the contestant's main-dish recipe?

Answer: ginger

Answer: lemongrass

Answer: nutmeg

Answer: saffron

Answer: turmeric


Question: If the condition that requires ginger to be included in the same recipe as nutmeg is suspended but all of the other original conditions remain in effect, then which one of the following could be a list of all of the flavorings included in one contestant's main-dish recipe?

Answer: ginger, lemongrass, nutmeg, paprika

Answer: ginger, lemongrass, paprika, turmeric

Answer: fenugreek, ginger, lemongrass, paprika, saffron

Answer: fenugreek, ginger, lemongrass, saffron, turmeric

Answer: fenugreek, lemongrass, nutmeg, paprika, saffron


Context: Seven singers—Jamie, Ken, Lalitha, Maya, Norton, Olive, and Patrick—will be scheduled to perform in the finals of a singing competition. During the evening of the competition, each singer, performing alone, will give exactly one performance. The schedule for the evening must conform to the following requirements: Jamie performs immediately after Ken. Patrick performs at some time after Maya. Lalitha performs third only if Norton performs fifth. If Patrick does not perform second, he performs fifth.

Question: Which one of the following is an acceptable schedule for the evening's performers, from first through seventh?

Answer: Ken, Jamie, Maya, Lalitha, Patrick, Norton, Olive

Answer: Lalitha, Patrick, Norton, Olive, Maya, Ken, Jamie

Answer: Norton, Olive, Ken, Jamie, Maya, Patrick, Lalitha

Answer: Olive, Maya, Ken, Lalitha, Patrick, Norton, Jamie

Answer: Olive, Maya, Lalitha, Norton, Patrick, Ken, Jamie


Question: If Lalitha is scheduled for the third performance, which one of the following must be scheduled for the sixth performance?

Answer: Jamie

Answer: Ken

Answer: Norton

Answer: Olive

Answer: Patrick


Question: If Norton is scheduled for the fifth performance, which one of the following could be true?

Answer: Jamie is scheduled for the sixth performance.

Answer: Ken is scheduled for the second performance.

Answer: Lalitha is scheduled for the fourth performance.

Answer: Maya is scheduled for the third performance.

Answer: Olive is scheduled for the first performance.


Question: If Maya is scheduled for the second performance, which one of the following could be true?

Answer: Jamie is scheduled for the sixth performance.

Answer: Ken is scheduled for the fourth performance.

Answer: Lalitha is scheduled for the third performance.

Answer: Norton is scheduled for the fifth performance.

Answer: Olive is scheduled for the fourth performance.


Question: If Jamie's performance is scheduled to be immediately before Lalitha's performance, Jamie's performance CANNOT be scheduled to be

Answer: second

Answer: third

Answer: fourth

Answer: fifth

Answer: sixth


Context: At a small press, six textbooks, three introductory—F, G, and H—and three advanced—X, Y, and Z—will each be evaluated once by the editor, Juarez, and once by the publisher, Rosenberg, during six consecutive weeks—week 1 through week 6. Each evaluator evaluates exactly one textbook per week. No textbook will be evaluated by Juarez and Rosenberg during the same week. The following additional constraints apply: Rosenberg cannot evaluate any introductory textbook until Juarez has evaluated that textbook. Juarez cannot evaluate any advanced textbook until Rosenberg has evaluated that textbook. Rosenberg cannot evaluate any two introductory textbooks consecutively. Juarez must evaluate X during week 4.

Question: Which one of the following is an acceptable evaluation schedule, with the textbooks listed in order of evaluation from week 1 through week 6?

Answer: Juarez: F, G, X, Z, H, Y Rosenberg: X, F, Z, G, Y, H

Answer: Juarez: F, Y, G, X, H, Z Rosenberg Y, F, X, G, Z, H

Answer: Juarez: G, H, F, X, Y, Z Rosenberg: X, G, H, Y, Z, F

Answer: Juarez: G, Z, F, X, H, Y Rosenberg: Z, F, X, G, Y, H

Answer: Juarez: H, Y, F, X, G, Z Rosenberg: X, H, Z, F, Y, G


Question: If Juarez evaluates H during week 3 and Rosenberg evaluates G during week 6, which one of the following must be true?

Answer: Juarez evaluates F during week 1.

Answer: Juarez evaluates G during week 2.

Answer: Juarez evaluates Z during week 6.

Answer: Rosenberg evaluates X during week 1.

Answer: Rosenberg evaluates Y during week 5.


Question: If Juarez evaluates Z during week 2, then Rosenberg must evaluate which one of the following textbooks during week 5?

Answer: F

Answer: H

Answer: X

Answer: Y

Answer: Z


Question: Which one of the following must be true?

Answer: Rosenberg evaluates H during week 6.

Answer: Rosenberg evaluates an advanced textbook during week 3.

Answer: Juarez evaluates an advanced textbook during week 2.

Answer: Juarez evaluates Y before evaluating G.

Answer: Juarez does not evaluate any two introductory textbooks consecutively.


Question: If Rosenberg evaluates X during week 1 and F during week 2, which one of the following could be true?

Answer: X is the third of the advanced textbooks to be evaluated by Juarez.

Answer: Y is the first of the advanced textbooks to be evaluated by Juarez.

Answer: Juarez does not evaluate any two introductory textbooks in a row.

Answer: Juarez evaluates G during week 5.

Answer: Juarez evaluates Z during week 6.


Question: Which one of the following is a complete and accurate list of those weeks during which Juarez must evaluate an introductory textbook?

Answer: week 1

Answer: week 6

Answer: week 1, week 5

Answer: week 1, week 2, week 3

Answer: week 1, week 3, week 5


Question: Which one of the following could be true?

Answer: Juarez evaluates F during week 6.

Answer: Juarez evaluates Z during week 1.

Answer: Rosenberg evaluates F during week 3.

Answer: Rosenberg evaluates H during week 2.

Answer: Rosenberg evaluates X during week 5.


Context: Nine different treatments are available for a certain illness: three antibiotics—F, G, and H—three dietary regimens—M, N, and O—and three physical therapies—U, V, and W. For each case of the illness, a doctor will prescribe exactly five of the treatments, in accordance with the following conditions: If two of the antibiotics are prescribed, the remaining antibiotic cannot be prescribed. There must be exactly one dietary regimen prescribed. If O is not prescribed, F cannot be prescribed. If W is prescribed, F cannot be prescribed. G cannot be prescribed if both N and U are prescribed. V cannot be prescribed unless both H and M are prescribed.

Question: Which one of the following could be the five treatments prescribed for a given case?

Answer: F, G, H, M, V

Answer: F, G, M, O, V

Answer: F, H, M, O, W

Answer: G, H, N, U, W

Answer: G, H, O, U, W


Question: Which one of the following could be the antibiotics and physical therapies prescribed for a given case?

Answer: F, G, H, W

Answer: F, G, U, V

Answer: F, U, V, W

Answer: G, U, V, W

Answer: H, U, V, W


Question: If O is prescribed for a given case, which one of the following is a pair of treatments both of which must also be prescribed for that case?

Answer: F, M

Answer: G, V

Answer: N, U

Answer: U, V

Answer: U, W


Question: If G is prescribed for a given case, which one of the following is a pair of treatments both of which could also be prescribed for that case?

Answer: F, M

Answer: F, N

Answer: N, V

Answer: O, V

Answer: V, W


Question: Which one of the following is a list of three treatments that could be prescribed together for a given case?

Answer: F, M, U

Answer: F, O, W

Answer: G, N, V

Answer: G, V, W

Answer: H, N, V


Question: Which one of the following treatments CANNOT be prescribed for any case?

Answer: G

Answer: M

Answer: N

Answer: U

Answer: W


Context: The members of two committees, a planting committee and a trails committee, are to be selected from among seven volunteers—F, G, H, J, K, L, and M. The following conditions govern the composition of the committees: Each committee must have at least three members. F cannot be on the same committee as K. If K is on a committee, J must also be on that committee. M must be on at least one of the committees. The two committees must have at least one member in common.

Question: Which one of the following represents an acceptable selection of volunteers for the committees?

Answer: planting: F, G, H; trails: G, J, K, L

Answer: planting: F, H, J; trails: G, H, L, M

Answer: planting: F, H, M; trails: G, K, L, M

Answer: planting: F, G, L, M; trails: F, H

Answer: planting: F, H, J, K; trails: H, L, M


Question: If the planting committee consists of F, H, L, and M, and if the trails committee consists of G, H, and J, then K could replace which one of the following committee members on a committee without violating any of the conditions governing the composition of the committees?

Answer: F

Answer: G

Answer: H

Answer: L

Answer: M


Question: If the only members of the planting committee are G, H, and L and if the two committees are to have as many members in common as the conditions allow, then which one of the following must be true?

Answer: The trails committee and the planting committee have exactly one member in common.

Answer: The trails committee and the planting committee have exactly two members in common.

Answer: The trails committee and the planting committee have an equal number of members.

Answer: The trails committee has at least one more member than the planting committee.

Answer: The planting committee has exactly two more members than the trails committee.


Question: If K is on both committees and L is also on both committees and if the planting committee has exactly three members, then which one of the following must be true?

Answer: F is on the planting committee.

Answer: F is on the trails committee.

Answer: G is on the planting committee.

Answer: M is on the planting committee.

Answer: M is on the trails committee.


Question: The largest number of members that the planting committee and the trails committee could have in common is

Answer: three

Answer: four

Answer: five

Answer: six

Answer: seven


Context: Exactly six tourists—Harry, Irene, Klaus, Laura, Michael, Norma—are to be assigned to four guides: Valois, Xerxes, Yossarian, Zalamea. Each tourist is assigned to exactly one guide, with at least one tourist assigned to each guide. Valois speaks only French. Xerxes speaks only Turkish and Spanish. Yossarian speaks only French and Turkish. Zalamea speaks only Spanish and Russian. Each tourist speaks exactly one of the languages spoken by his or her guide and speaks no other language. The following rules govern the assignment of the tourists to the guides: At least Harry and Irene are assigned to Yossarian. At least Laura is assigned to Zalamea. If Klaus is assigned to Xerxes, then Michael speaks French.

Question: Each of the following could be true EXCEPT:

Answer: Both Klaus and Harry speak Turkish.

Answer: Both Klaus and Michael speak French.

Answer: Both Klaus and Michael speak Russian.

Answer: Both Klaus and Norma speak French.

Answer: Both Klaus and Norma speak Spanish.


Question: Which one of the following must be true?

Answer: Zalamea is assigned fewer than three of the tourists.

Answer: Xerxes is assigned fewer than two of the tourists.

Answer: Yossarian is assigned exactly two of the tourists.

Answer: Valois is assigned exactly one of the tourists.

Answer: Zalamea is assigned exactly one of the tourists.


Question: Each of the following could be true of the assignment of tourists to guides EXCEPT:

Answer: It assigns Klaus to Valois and Michael to Xerxes.

Answer: It assigns Klaus to Yossarian and Norma to Zalamea.

Answer: It assigns Laura to Zalamea and Michael to Zalamea.

Answer: It assigns Michael to Valois and Klaus to Zalamea.

Answer: It assigns Michael to Xerxes and Klaus to Zalamea.


Question: If Klaus and Laura speak the same language as each other, then which one of the following must be true?

Answer: At least one of Michael and Norma speaks Spanish.

Answer: At least two tourists speak Russian.

Answer: Klaus and Laura speak Russian.

Answer: At least two tourists speak French.

Answer: At least one of Michael and Norma speaks French.


Question: If Laura and Norma speak the same language as each other, then the maximum number of the tourists who could speak Turkish is

Answer: two

Answer: three

Answer: four

Answer: five

Answer: six


Question: If exactly two tourists are assigned to Xerxes, then which one of the following could be true?

Answer: Norma speaks Russian and Laura speaks Russian.

Answer: Norma speaks French and Michael speaks French.

Answer: Norma speaks French and Klaus speaks Turkish.

Answer: Michael speaks Spanish and Klaus speaks Spanish.

Answer: Michael speaks French and Klaus speaks Spanish.


Question: If Harry, Irene, Michael, and Norma all speak the same language as each other, then which one of the following could be true?

Answer: Klaus speaks Russian.

Answer: Exactly two of the tourists speak Russian.

Answer: Exactly three of the tourists speak Spanish.

Answer: Exactly two of the tourists speak Turkish.

Answer: Klaus speaks French.


Context: Each of six people—Kim, Lina, Maricella, Oliver, Paulo, and Shigeru—plays exactly one of two sports—golf, and tennis. If a sport is played by more than one of the six people, then the people who play that sport are ranked in order of ability from highest to lowest, with no ties. The following conditions must apply: Oliver plays tennis. Lina plays golf. There is no golf player ranked higher than Lina. If Maricella plays golf, then Paulo and Shigeru play golf, Paulo ranking lower than Maricella but higher than Shigeru. If Maricella plays tennis, then Shigeru plays tennis, Shigeru ranking lower than Oliver but higher than Maricella. If Paulo plays tennis, then Kim plays tennis, Oliver ranking lower than Kim but higher than Paulo.

Question: Which one of the following could be true?

Answer: Maricella plays tennis and Shigeru plays golf.

Answer: Paulo plays tennis and Kim plays golf.

Answer: Kim plays tennis and Paulo plays golf.

Answer: Paulo and Oliver play tennis, Paulo ranking higher than Oliver.

Answer: Maricella and Shigeru play tennis, Maricella ranking higher than Shigeru.


Question: Each of the following could be the lowest-ranking tennis player EXCEPT:

Answer: Shigeru

Answer: Kim

Answer: Oliver

Answer: Paulo

Answer: Maricella


Question: Which one of the following could be a complete and accurate list of the people who play tennis, listed in order of rank from highest to lowest?

Answer: Shigeru, Paulo, Oliver

Answer: Oliver, Paulo, Kim

Answer: Shigeru, Oliver, Maricella

Answer: Oliver, Shigeru, Kim, Maricella

Answer: Paulo, Maricella, Shigeru, Kim


Question: If Shigeru plays golf, then each of the following are pairs of people who must play the same sport as each other EXCEPT:

Answer: Paulo and Maricella

Answer: Paulo and Shigeru

Answer: Kim and Maricella

Answer: Paulo and Lina

Answer: Maricella and Shigeru


Question: If Oliver is the highest-ranking tennis player, which one of the following must be true?

Answer: Oliver and Maricella play the same sport as each other.

Answer: Paulo and Lina play the same sport as each other.

Answer: Paulo and Oliver play the same sport as each other.

Answer: Kim and Lina do not play the same sport as each other.

Answer: Kim and Paulo do not play the same sport as each other.


Question: Suppose that the condition is added that Shigeru and Paulo do not play the same sport as each other. If all the other conditions remain in effect, then each of the following could be true EXCEPT:

Answer: Maricella and Kim play the same sport as each other.

Answer: Paulo and Kim play the same sport as each other.

Answer: Paulo and Maricella play the same sport as each other.

Answer: Kim and Paulo do not play the same sport as each other.

Answer: Maricella and Kim do not play the same sport as each other.


Context: A disc jockey will play a sequence consisting of exactly seven different songs: three ballads—F, G, and H—and four dance tunes—R, S, V, and X. The following conditions must be met: No dance tune can be played immediately after another dance tune. H must be played earlier in the sequence than V. V and S must be separated from each other by exactly one song. S must be played immediately before or immediately after F. F must be played immediately after R, unless G is played earlier than R.

Question: Which one of the following could be the order of the songs in the sequence?

Answer: G, H, S, X, V, F, R

Answer: R, H, X, G, S, F, V

Answer: S, F, X, G, R, H, V

Answer: V, F, S, H, X, G, R

Answer: X, G, R, H, S, F, V


Question: Which one of the following must be true about the sequence?

Answer: The first song is X.

Answer: The fifth song is S.

Answer: No ballad is played immediately after a dance tune.

Answer: No ballad is played immediately after another ballad.

Answer: No dance tune is played immediately after a ballad.


Question: Which one of the following could be the fourth song in the sequence?

Answer: G

Answer: R

Answer: S

Answer: V

Answer: X


Question: Which one of the following could be the first song in the sequence?

Answer: R

Answer: S

Answer: V

Answer: F

Answer: G


Question: If the third song in the sequence is S, which one of the following must be the sixth song?

Answer: G

Answer: H

Answer: R

Answer: V

Answer: X


Question: If the seventh song in the sequence is R, which one of the following could be the fifth song?

Answer: F

Answer: G

Answer: H

Answer: V

Answer: X


Context: Eight physics students—four majors: Frank, Gwen, Henry, and Joan; and four nonmajors: Victor, Wanda, Xavier, and Yvette—are being assigned to four laboratory benches, numbered 1 through 4. Each student is assigned to exactly one bench, and exactly two students are assigned to each bench. Assignments of students to benches must conform to the following conditions: Exactly one major is assigned to each bench. Frank and Joan are assigned to consecutively numbered benches, with Frank assigned to the lower-numbered bench. Frank is assigned to the same bench as Victor. Gwen is not assigned to the same bench as Wanda.

Question: Which one of the following could be the assignment of students to benches?

Answer: 1: Frank, Victor; 2: Joan, Gwen; 3: Henry, Wanda; 4: Xavier, Yvette

Answer: 1: Gwen, Yvette; 2: Frank, Xavier; 3: Joan, Wanda; 4: Henry, Victor

Answer: 1: Henry, Wanda; 2: Gwen, Xavier; 3: Frank, Victor; 4: Joan, Yvette

Answer: 1: Henry, Xavier; 2: Joan, Wanda; 3: Frank, Victor; 4: Gwen, Yvette

Answer: 1: Henry, Yvette; 2: Gwen, Wanda; 3: Frank, Victor, 4: Joan, Xavier


Question: If Victor is assigned to bench 2 and Wanda is assigned to bench 4, which one of the following must be true?

Answer: Frank is assigned to bench 1.

Answer: Gwen is assigned to bench 1.

Answer: Henry is assigned to bench 3.

Answer: Xavier is assigned to bench 1.

Answer: Yvette is assigned to bench 3.


Question: If Gwen and Henry are not assigned to consecutively numbered benches, which one of the following must be true?

Answer: Victor is assigned to bench 2.

Answer: Victor is assigned to bench 3.

Answer: Wanda is assigned to bench 1.

Answer: Wanda is assigned to bench 3.

Answer: Wanda is assigned to bench 4.


Question: If Henry and Yvette are both assigned to bench 1, which one of the following could be true?

Answer: Gwen is assigned to bench 3.

Answer: Joan is assigned to bench 2.

Answer: Wanda is assigned to bench 2.

Answer: Wanda is assigned to bench 3.

Answer: Xavier is assigned to bench 3.


Question: If Gwen is assigned to bench 4 and Xavier is assigned to bench 3, then any one of the following could be true EXCEPT:

Answer: Gwen is assigned to the same bench as Yvette.

Answer: Henry is assigned to the same bench as Wanda.

Answer: Henry is assigned to the same bench as Xavier.

Answer: Joan is assigned to the same bench as Xavier.

Answer: Joan is assigned to the same bench as Yvette.


Question: If Wanda is assigned to a lower-numbered bench than is Joan, then Henry must be assigned to a

Answer: lower-numbered bench than is Frank

Answer: lower-numbered bench than is Gwen

Answer: lower-numbered bench than is Xavier

Answer: higher-numbered bench than is Victor

Answer: higher-numbered bench than is Yvette


Question: Which one of the following could be the assignments for bench 2 and bench 4?

Answer: 2: Gwen, Xavier 4: Henry, Yvette

Answer: 2: Henry, Yvette 4: Joan, Xavier

Answer: 2: Joan, Victor 4: Gwen, Xavier

Answer: 2: Joan, Wanda 4: Gwen, Xavier

Answer: 2: Joan, Xavier 4: Henry, Yvette


Context: A messenger will deliver exactly seven packages—L, M, N, O, P, S, and T—one at a time, not necessarily in that order. The seven deliveries must be made according to the following conditions: P is delivered either first or seventh. The messenger delivers N at some time after delivering L. The messenger delivers T at some time after delivering M. The messenger delivers exactly one package between delivering L and delivering O, whether or not L is delivered before O. The messenger delivers exactly one package between delivering M and delivering P, whether or not M is delivered before P.

Question: Which one of the following is an order in which the messenger could make the deliveries, from first to seventh?

Answer: L, N, S, O, M, T, P

Answer: M, T, P, S, L, N, O

Answer: O, S, L, N, M, T, P

Answer: P, N, M, S, O, T, L

Answer: P, T, M, S, L, N, O


Question: Which one of the following could be true?

Answer: N is delivered first.

Answer: T is delivered first.

Answer: T is delivered second.

Answer: M is delivered fourth.

Answer: S is delivered seventh.


Question: If N is delivered fourth, which one of the following could be true?

Answer: L is delivered first.

Answer: L is delivered second.

Answer: M is delivered third.

Answer: O is delivered fifth.

Answer: S is delivered first.


Question: If T is delivered fourth, the seventh package delivered must be

Answer: L

Answer: N

Answer: O

Answer: P

Answer: S


Question: If the messenger delivers M at some time after delivering O, the fifth package delivered could be any one of the following EXCEPT:

Answer: L

Answer: M

Answer: N

Answer: S

Answer: T


Context: Each of exactly five persons—Nguyen, Olson, Pike, Tyner, and Valdez—participates in exactly one of three activities: going to a movie, going to a soccer game, or going to a restaurant. The following conditions must apply. Nguyen and Olson do not participate in the same activity as each other, nor does either one of them participate in the same activity as Pike. Exactly two persons go to a soccer game. Tyner and Pike do not participate in the same activity as each other. If Nguyen or Valdez goes to a movie, they both go to a movie.

Question: Which one of the following could be an accurate list of the activities participated in by Nguyen, Olson, Pike, Tyner, and Valdez, respectively?

Answer: movie, soccer game, soccer game, restaurant, movie

Answer: movie, restaurant, soccer game, soccer game, movie

Answer: soccer game, restaurant, movie, soccer game, movie

Answer: soccer game, restaurant, movie, soccer game, restaurant

Answer: soccer game, restaurant, movie, soccer game, soccer game


Question: If Valdez goes to a soccer game, then each of the following could be true EXCEPT:

Answer: Olson goes to a movie.

Answer: Nguyen goes to a restaurant.

Answer: Nguyen goes to a soccer game.

Answer: Tyner goes to a soccer game.

Answer: Tyner goes to a movie.


Question: Which one of the following is a pair of persons who could go to a movie together?

Answer: Nguyen and Tyner

Answer: Olson and Tyner

Answer: Olson and Valdez

Answer: Pike and Olson

Answer: Pike and Tyner


Question: Each of the following statements must be false EXCEPT:

Answer: Only Olson goes to a restaurant.

Answer: Only Pike goes to a restaurant.

Answer: Only Tyner goes to a restaurant.

Answer: Only Valdez goes to a restaurant.

Answer: Tyner and Valdez go to a restaurant together.


Question: If Nguyen goes to a soccer game, then which one of the following is a complete and accurate list of the persons any one of whom could go to a movie?

Answer: Olson

Answer: Pike, Valdez

Answer: Olson, Tyner

Answer: Pike, Tyner, Valdez

Answer: Olson, Pike, Tyner


Question: If the condition that exactly two persons go to a soccer game is changed to require that exactly three persons go to a soccer game, but all other conditions remain the same, then which one of the following persons must participate in an activity other than going to a soccer game?

Answer: Nguyen

Answer: Olson

Answer: Pike

Answer: Tyner

Answer: Valdez


Context: In each of two years exactly two of four lawmakers—Feld, Gibson, Hsu, and Ivins—and exactly two of three scientists—Vega, Young, and Zapora—will serve as members of a four-person panel. In each year, one of the members will be chairperson. The chairperson in the first year cannot serve on the panel in the second year. The chairperson in the second year must have served on the panel in the first year. Service on the panel must obey the following conditions: Gibson and Vega do not serve on the panel in the same year as each other. Hsu and Young do not serve on the panel in the same year as each other. Each year, either Ivins or Vega, but not both, serves on the panel.

Question: Which one of the following could be the list of the people who serve on the panel in the first year?

Answer: Feld, Gibson, Vega, Zapora

Answer: Feld, Hsu, Vega, Zapora

Answer: Feld, Ivins, Vega, Zapora

Answer: Gibson, Hsu, Ivins, Zapora

Answer: Hsu, Ivins, Young, Zapora


Question: If Vega is the chairperson in the first year, which one of the following is a pair of people who must serve on the panel in the second year?

Answer: Gibson and Young

Answer: Gibson and Zapora

Answer: Hsu and Ivins

Answer: Ivins and Young

Answer: Vega and Young


Question: If Hsu is the chairperson in the first year, which one of the following could be the chairperson in the second year?

Answer: Feld

Answer: Gibson

Answer: Hsu

Answer: Ivins

Answer: Young


Question: IF Feld serves on the panel in a given year, any one of the following could serve on the panel that year EXCEPT:

Answer: Gibson

Answer: Hsu

Answer: Ivins

Answer: Vega

Answer: Young


Question: If Ivins is the chairperson in the first year, which one of the following could be the chairperson in the second year?

Answer: Feld

Answer: Gibson

Answer: Hsu

Answer: Vega

Answer: Young


Question: Which one of the following must be true?

Answer: Feld is on the panel in the second year.

Answer: Hsu is on the panel in the first year.

Answer: Ivins is on the panel in both years.

Answer: Young is on the panel in both years.

Answer: Zapora is on the panel in the second year.


Context: During a period of seven consecutive days—from day 1 through day 7—seven investors—Fennelly, Gupta, Hall, Jones, Knight, Lopez, and Moss—will each view a building site exactly once. Each day exactly one investor will view the site. The investors must view the site in accordance with the following conditions: Fennelly views the site on day 3 or else day 5. Lopez views the site on neither day 4 nor day 6. If Jones views the site on day 1, Hall views the site on day 2. If Knight views the site on day 4, Lopez views the site on day 5. Gupta views the site on the day after the day on which Hall views the site.

Question: Which one of the following could be the order in which the investors view the site, from day 1 through day 7?

Answer: Hall, Gupta, Fennelly, Moss, Knight, Lopez, Jones

Answer: Hall, Gupta, Lopez, Fennelly, Moss, Knight, Jones

Answer: Lopez, Gupta, Hall, Moss, Fennelly, Jones, Knight

Answer: Lopez, Jones, Fennelly, Knight, Hall, Gupta, Moss

Answer: Lopez, Jones, Knight, Moss, Fennelly, Hall, Gupta


Question: If Jones views the site on day 1, which one of the following investors must view the site on day 4?

Answer: Fennelly

Answer: Gupta

Answer: Knight

Answer: Lopez

Answer: Moss


Question: If Knight views the site on day 4 and Moss views the site on some day after the day on which Jones views the site, which one of the following must be true?

Answer: Jones views the site on day 1.

Answer: Jones views the site on day 2.

Answer: Jones views the site on day 6.

Answer: Moss views the site on day 2.

Answer: Moss views the site on day 6.


Question: If Hall views the site on day 2, which one of the following is a complete and accurate list of investors any one of whom could be the investor who views the site on day 4?

Answer: Knight

Answer: Moss

Answer: Jones, Moss

Answer: Knight, Moss

Answer: Jones, Knight, Moss


Question: If Hall views the site on the day after the day Knight views the site and if Fennelly views the site on the day after the day Lopez views the site, then Jones must view the site on day

Answer: 1

Answer: 2

Answer: 3

Answer: 4

Answer: 5


Question: If the day on which Gupta views the site and the day on which Lopez views the site both come at some time before the day on which Fennelly views the site, which one of the following is an investor who could view the site on day 3?

Answer: Fennelly

Answer: Gupta

Answer: Jones

Answer: Knight

Answer: Moss


Context: A zoo's reptile house has a straight row of exactly five consecutive habitats—numbered 1 through 5 from left to right—for housing exactly seven reptiles—four snakes and three lizards. Five of the reptiles are female and two are male. The reptiles must be housed as follows: No habitat houses more than two reptiles. No habitat houses both a snake and a lizard. No female snake is housed in a habitat that is immediately next to a habitat housing a male lizard.

Question: Which one of the following could be a complete and accurate matching of habitats to reptiles?

Answer: 1: two female snakes; 2: one male snake; 3: one female lizard; 4: one male snake, one female lizard; 5: one female lizard

Answer: 1: empty; 2: two female snakes; 3: two female lizards; 4: two male snakes; 5: one female lizard

Answer: 1: one female snake, one male snake; 2: two female snakes; 3: one male lizard; 4: one female lizard; 5: one female lizard

Answer: 1: two male snakes; 2: empty; 3: one female lizard; 4: one female lizard; 5: two female snakes, one female lizard

Answer: 1: one female snake, one male snake; 2: one female snake, one male snake; 3: one male lizard; 4: one female lizard; 5: one female lizard


Question: If habitat 2 contains at least one female snake and habitat 4 contains two male lizards, then which one of the following could be true?

Answer: Habitat 3 contains two reptiles.

Answer: Habitat 5 contains two reptiles.

Answer: Habitat 1 contains a female lizard.

Answer: Habitat 2 contains a female lizard.

Answer: Habitat 5 contains a female lizard.


Question: Which one of the following must be true?

Answer: At least one female reptile is alone in a habitat.

Answer: At least one male reptile is alone in a habitat.

Answer: At least one lizard is alone in a habitat.

Answer: At least one lizard is male.

Answer: At least one snake is male.


Question: Which one of the following CANNOT be the complete housing arrangement for habitats 1 and 2?

Answer: 1: one female snake, one male snake; 2: one male snake

Answer: 1: one male lizard; 2: one male snake

Answer: 1: two female lizards; 2: one female snake

Answer: 1: one male snake; 2: empty

Answer: 1: empty; 2: one female lizard


Question: If habitat 3 is empty, and no snake is housed in a habitat that is immediately next to a habitat containing a snake, then which one of the following could be false?

Answer: All snakes are housed in even-numbered habitats.

Answer: None of the lizards is male.

Answer: No snake is alone in a habitat.

Answer: No lizard is housed in a habitat that is immediately next to a habitat containing a lizard.

Answer: Exactly one habitat contains exactly one reptile.


Question: If all snakes are female and each of the lizards has a habitat to itself, then which one of the following habitats CANNOT contain any snakes?

Answer: habitat 1

Answer: habitat 2

Answer: habitat 3

Answer: habitat 4

Answer: habitat 5


Context: Exactly seven film buffs—Ginnie, Ian, Lianna, Marcos, Reveka, Viktor, and Yow—attend a showing of classic films. Three films are shown, one directed by Fellini, one by Hitchcock, and one by Kurosawa. Each of the film buffs sees exactly one of the three films. The films are shown only once, one film at a time. The following restrictions must apply: Exactly twice as many of the film buffs see the Hitchcock film as see the Fellini film. Ginnie and Reveka do not see the same film as each other. Ian and Marcos do not see the same film as each other. Viktor and Yow see the same film as each other. Lianna sees the Hitchcock film. Ginnie sees either the Fellini film or the Kurosawa film.

Question: Which one of the following could be an accurate matching of film buffs to films?

Answer: Ginnie: the Hitchcock film; Ian: the Kurosawa film; Marcos: the Hitchcock film

Answer: Ginnie: the Kurosawa film; Ian: the Fellini film; Viktor: the Fellini film

Answer: Ian: the Hitchcock film; Reveka: the Kurosawa film; Viktor: the Fellini film

Answer: Marcos: the Kurosawa film; Reveka: the Kurosawa film; Viktor: the Kurosawa film

Answer: Marcos: the Hitchcock film; Reveka: the Hitchcock film; Yow: the Hitchcock film


Question: Each of the following must be false EXCEPT:

Answer: Reveka is the only film buff to see the Fellini film.

Answer: Reveka is the only film buff to see the Hitchcock film.

Answer: Yow is the only film buff to see the Kurosawa film.

Answer: Exactly two film buffs see the Kurosawa film.

Answer: Exactly three film buffs see the Hitchcock film.


Question: Which one of the following could be a complete and accurate list of the film buffs who do NOT see the Hitchcock film?

Answer: Ginnie, Marcos

Answer: Ginnie Reveka

Answer: Ginnie, Ian, Reveka

Answer: Ginnie, Marcos, Yow

Answer: Ginnie, Viktor, Yow


Question: If exactly one film buff sees the Kurosawa film, then which one of the following must be true?

Answer: Viktor sees the Hitchcock film.

Answer: Ginnie sees the Fellini film.

Answer: Marcos sees the Fellini film.

Answer: Ian sees the Fellini film.

Answer: Reveka sees the Hitchcock film.


Question: Which one of the following must be true?

Answer: Ginnie sees a different film than Ian does.

Answer: Ian sees a different film than Lianna does.

Answer: Ian sees a different film than Viktor does.

Answer: Ian, Lianna, and Viktor do not all see the same film.

Answer: Ginnie, Lianna, and Marcos do not all see the same film.


Question: If Viktor sees the same film as Ginnie does, then which one of the following could be true?

Answer: Ginnie sees the Fellini film.

Answer: Ian sees the Hitchcock film.

Answer: Reveka sees the Kurosawa film.

Answer: Viktor sees the Hitchcock film.

Answer: Yow sees the Fellini film.


Question: Each of the following could be complete and accurate list of the film buffs who see the Fellini film EXCEPT:

Answer: Ginnie, Ian

Answer: Ginnie, Marcos

Answer: Ian, Reveka

Answer: Marcos, Reveka

Answer: Viktor, Yow


Context: Six cars are to be arranged in a straight line, and will be numbered 1 through 6, in order, from the front of the line to the back of the line. Each car is exactly one color: two are green, two are orange, and two are purple. The arrangement of cars is restricted as follows: No car can be the same color as any car next to it in line. Either car 5 or car 6 must be purple. Car 1 cannot be orange. Car 4 cannot be green.

Question: The cars in which one of the following pairs CANNOT be the same color as each other?

Answer: cars 1 and 4

Answer: cars 1 and 5

Answer: cars 3 and 5

Answer: cars 3 and 6

Answer: cars 4 and 6


Question: If car 2 is the same color as car 4, then which one of the following statements must be true?

Answer: Car 1 is purple.

Answer: Car 2 is orange.

Answer: Car 3 is green.

Answer: Car 5 is purple.

Answer: Car 6 is green


Question: If car 4 is purple, which one of the following must be true?

Answer: Car 1 is orange.

Answer: Car 2 is green.

Answer: Car 3 is orange.

Answer: Car 5 is green.

Answer: Car 6 is purple.


Question: Which one of the following statements must be false?

Answer: Car 2 is green.

Answer: Car 4 is orange.

Answer: Car 5 is purple.

Answer: Car 6 is orange.

Answer: Car 6 is green.


Question: If one of the two orange cars is replaced by a third green car, and if the arrangement of cars in line must conform to the same restrictions as before, then which one of the following is a complete and accurate list of the cars each of which must be green?

Answer: car 1

Answer: car 3

Answer: car 5

Answer: car 1, car 3

Answer: car l, car 3, car 5


Context: Exactly eight boats—Jewel, Kashmir, Neptune, Ojibwa, Pacific, Spain, Tornado, and Valhalla—arrived at a dock. No boat arrived at the same time as any other boat. The boats arrived in an order consistent with the following conditions: Tornado arrived before Spain but after Jewel. Neptune arrived before Tornado. Kashmir arrived after Jewel but before Spain. Spain arrived before Ojibwa.

Question: If Neptune arrived after Kashmir, which one of the following must be false?

Answer: Jewel was the second of the boats to arrive.

Answer: Kashmir was the fifth of the boats to arrive.

Answer: Neptune was the third of the boats to arrive.

Answer: Ojibwa was the sixth of the boats to arrive.

Answer: Spain was the seventh of the boats to arrive.


Question: Which one of the following must be true?

Answer: At least two of the boats arrived before Neptune.

Answer: At least five of the boats arrived before Pacific.

Answer: At least four of the boats arrived before Spain.

Answer: At least three of the boats arrived before Tornado.

Answer: At least two of the boats arrived before Valhalla.


Question: Of the eight boats, what is the maximum number that could have arrived before Jewel?

Answer: none

Answer: one

Answer: two

Answer: three

Answer: four


Question: Of the eight boats, if Valhalla was the second to arrive, then which one of the following CANNOT be true?

Answer: Jewel was the third to arrive.

Answer: Jewel was the first to arrive.

Answer: Kashmir was the third to arrive.

Answer: Pacific was the third to arrive.

Answer: Tornado was the third to arrive.


Question: If Valhalla arrived before Neptune but after Pacific, which one of the following could be true?

Answer: Tornado arrived before Valhalla.

Answer: Kashmir arrived before Pacific.

Answer: Ojibwa was not the last of the boats to arrive.

Answer: Spain arrived before Valhalla.

Answer: Spain was not the seventh of the boats to arrive.


Question: Of the boats, what are, respectively, the minimum number and the maximum number that could have arrived before Kashmir?

Answer: one, five

Answer: two, five

Answer: three, five

Answer: one, six

Answer: two, six


Context: A park contains at most five of seven kinds of trees—firs, laurels, maples, oaks, pines, spruces, and yews—consistent with the following conditions: If maples are in the park, yews are not. If firs are in the park, pines are not. If yews are not in the park, then either laurels or oaks, but not both, are in the park. If it is not the case that the park contains both laurels and oaks, then it contains firs and spruces.

Question: Which one of the following could be a complete and accurate list of the kinds of trees in the park?

Answer: firs, maples

Answer: firs, laurels, oaks

Answer: firs, laurels, pines, spruces

Answer: firs, laurels, spruces, yews

Answer: firs, maples, oaks, spruces, yews


Question: If neither maples nor spruces are in the park, then which one of the following could be true?

Answer: Exactly four kinds of trees are in the park.

Answer: Exactly five kinds of trees are in the park.

Answer: Laurels are not in the park.

Answer: Oaks are not in the park.

Answer: Yews are not in the park.


Question: Which one of the following could be true?

Answer: Neither firs nor laurels are in the park.

Answer: Neither laurels, oaks, nor yews are in the park.

Answer: Neither laurels nor spruces are in the park.

Answer: Neither maples nor yews are in the park.

Answer: Neither oaks nor spruces are in the park.


Question: If firs are not in the park, then which one of the following must be true?

Answer: Maples are not in the park.

Answer: Spruces are not in the park.

Answer: Yews are not in the park.

Answer: Pines are in the park.

Answer: Spruces are in the park.


Question: If pines are in the park, then which one of the following must be true?

Answer: Exactly four kinds of trees are in the park.

Answer: Exactly five kinds of trees are in the park.

Answer: Neither firs nor maples are in the park.

Answer: Neither firs nor oaks are in the park.

Answer: Neither laurels nor maples are in the park.


Question: Each of the following could be an accurate, partial list of the kinds of trees in the park EXCEPT:

Answer: oaks, spruces

Answer: oaks, yews

Answer: firs, laurels, oaks

Answer: firs, maples, oaks

Answer: laurels, maples, oaks


Context: Four married couples—Francisco and Gabrielle, Kyoko and Lee, Olivia and Peter, Raymond and Simone—will dine together at the same circular table. Each person will sit in a different one of the eight chairs evenly spaced around the table. The chairs are numbered from 1 through 8 with successively numbered chairs next to each other and chair 1 next to chair 8. Seating arrangements must meet the following conditions: No person sits next to her or his spouse. Simone sits in chair 1 and Raymond sits in chair 4. Neither Peter nor Olivia sits next to Raymond. Neither Kyoko nor Lee sits next to Gabrielle.

Question: Which one of the following could be the list of people sitting in chairs 5 through 8, respectively?

Answer: Francisco, Olivia, Gabrielle, Peter

Answer: Francisco, Gabrielle, Peter, Lee

Answer: Olivia, Gabrielle, Peter, Kyoko

Answer: Gabrielle, Lee, Olivia, Kyoko

Answer: Lee, Olivia, Francisco, Kyoko


Question: Each of the following could be true EXCEPT:

Answer: Peter and Gabrielle sit in chairs 2 and 3, respectively.

Answer: Peter and Kyoko sit in chairs 1 and 3, respectively.

Answer: Francisco, Kyoko, and Peter sit in chairs 5, 6, and 7, respectively.

Answer: Gabrielle, Olivia, and Kyoko sit in chairs 5, 6, and 7, respectively.

Answer: Gabrielle, Peter, and Lee sit in chairs 5, 6, and 7, respectively.


Question: If Francisco sits in chair 2 and Olivia sits next to Simone, Gabrielle must sit next to which one of the following persons?

Answer: Francisco

Answer: Olivia

Answer: Lee

Answer: Raymond

Answer: Peter


Question: If neither Gabrielle nor Lee sits next to Raymond, which one of the following must be true?

Answer: Francisco sits next to Raymond.

Answer: Olivia sits next to Simone.

Answer: Francisco sits in chair 3.

Answer: Olivia sits in chair 6.

Answer: Olivia sits in chair 7.


Question: If Lee sits in chair 2, which one of the following must be true?

Answer: Peter sits next to Gabrielle.

Answer: Kyoko sits next to Simone.

Answer: Francisco sits next to Lee.

Answer: Gabrielle sits in chair 5.

Answer: Olivia sits in chair 6.


Question: If both Francisco and Gabrielle sit next to Raymond, which one of the following is a complete and accurate list of people any one of whom could sit next to Olivia?

Answer: Francisco, Gabrielle, Kyoko

Answer: Lee, Simone, Kyoko

Answer: Lee, Gabrielle, Kyoko

Answer: Francisco, Lee, Gabrielle, Kyoko

Answer: Lee, Simone, Gabrielle, Kyoko


Context: Zeno's Unfinished Furniture sells exactly five types of furniture—footstools, hutches, sideboards, tables, and vanities. Irene buys just four items, each of a different type, and each made entirely of one kind of wood—maple, oak, pine, or rosewood. The following conditions govern Irene's purchases: Any vanity she buys is maple. Any rosewood item she buys is a sideboard. If she buys a vanity, she does not buy a footstool. If Irene buys a footstool, she also buys a table made of the same wood. Irene does not buy an oak table. Exactly two of the items she buys are made of the same kind of wood as each other.

Question: Which one of the following could be an accurate list of the items Irene buys?

Answer: Maple footstool, maple hutch, rosewood sideboard, maple table

Answer: oak hutch, rosewood sideboard, pine table, oak vanity

Answer: rosewood hutch, maple sideboard, oak table, maple vanity

Answer: pine footstool, rosewood sideboard, pine table, maple vanity

Answer: maple footstool, pine hutch, oak sideboard, maple table


Question: If Irene buys one item made of rosewood and two items made of maple, then which one of the following pairs could be two of the items she buys?

Answer: a rosewood sideboard and an oak footstool

Answer: an oak hutch and a pine sideboard

Answer: an oak hutch and a maple table

Answer: a maple sideboard and a maple vanity

Answer: a maple hutch and a maple table


Question: Which one of the following is a complete and accurate list of all the woods any footstool that Irene buys could be made of?

Answer: maple, oak

Answer: maple, pine

Answer: maple, rosewood

Answer: maple, oak, pine

Answer: maple, oak, pine, rosewood


Question: Suppose Irene buys a footstool. Then which one of the following is a complete and accurate list of items any one of which she could buy in maple?

Answer: footstool, hutch, sideboard, table, vanity

Answer: footstool, hutch, sideboard, table

Answer: footstool, hutch, sideboard

Answer: footstool, hutch

Answer: footstool


Question: Which one of the following CANNOT be the two items Irene buys that are made of the same wood as each other?

Answer: footstool, hutch

Answer: hutch, sideboard

Answer: hutch, table

Answer: sideboard, vanity

Answer: table, vanity


Question: If Irene does not buy an item made of maple, then each of the following must be true EXCEPT:

Answer: Irene buys a footstool.

Answer: Irene buys a pine hutch.

Answer: Irene buys a rosewood sideboard.

Answer: Irene buys exactly one item made of oak.

Answer: Irene buys exactly two items made of pine.


Context: Six racehorses—K, L, M, N, O, and P—will be assigned to six positions arranged in a straight line and numbered consecutively 1 through 6. The horses are assigned to the positions, one horse per position, according to the following conditions: K and L must be assigned to positions that are separated from each other by exactly one position. K and N cannot be assigned to positions that are next to each other. N must be assigned to a higher-numbered position than M. P must be assigned to position 3.

Question: Which one of the following lists an acceptable assignment of horses to positions 1 though 6, respectively?

Answer: K, L, P, M, N, O

Answer: M, K, P, L, N, O

Answer: M, N, K, P, L, O

Answer: N, O, P, K, M, L

Answer: O, M, P, L, N, K


Question: Which one of the following is a complete and accurate list of the positions any one of which can be the position to which K is assigned?

Answer: 1, 2

Answer: 2, 3

Answer: 2, 4

Answer: 2, 4, 5

Answer: 2, 4, 6


Question: Which one of the following CANNOT be true?

Answer: K is assigned to position 2.

Answer: L is assigned to position 2.

Answer: M is assigned to position 1.

Answer: M is assigned to position 5.

Answer: O is assigned to position 2.


Question: Which one of the following must be true?

Answer: Either K or else L is assigned to position 2.

Answer: Either K or else L is assigned to position 4.

Answer: Either M or else N is assigned to position 2.

Answer: Either M or else N is assigned to position 5.

Answer: Either M or else O is assigned to position 6.


Question: Which one of the following CANNOT be true?

Answer: L and N are assigned to positions that are next to each other.

Answer: M and K are assigned to positions that are next to each other.

Answer: M and O are assigned to positions that are next to each other.

Answer: L and N are assigned to positions that are separated from each other by exactly one position.

Answer: M and P are assigned to positions that are separated from each other by exactly one position.


Context: To prepare for fieldwork, exactly four different researchers—a geologist, a historian, a linguist, and a paleontologist—will learn at least one and at most three of four languages—Rundi, Swahili, Tigrinya, and Yoruba. They must learn the languages according to the following specifications: Exactly one researcher learns Rundi. Exactly two researchers learn Swahili. Exactly two researchers learn Tigrinya. Exactly three researchers learn Yoruba. Any language learned by the linguist or paleontologist is not learned by the geologist. Any language learned by the geologist is learned by the historian.

Question: Which one of the following could be true?

Answer: The linguist learns three languages—Rundi, Swahili, and Tigrinya.

Answer: The linguist learns three languages—Swahili, Tigrinya, and Yoruba.

Answer: The historian learns three languages—Rundi, Swahili, and Tigrinya.

Answer: The historian learns three languages—Swahili, Tigrinya, and Yoruba.

Answer: The paleontologist learns three languages—Rundi, Swahili, and Tigrinya.


Question: If the linguist learns three of the languages, then which one of the following must be true?

Answer: The linguist learns Tigrinya.

Answer: The linguist learns Rundi.

Answer: The linguist learns Swahili.

Answer: The paleontologist learns Rundi.

Answer: The paleontologist learns Swahili.


Question: Each of the following could be true of the researcher who learns Rundi EXCEPT:

Answer: The researcher also learns Tigrinya but not Swahili.

Answer: The researcher learns neither Tigrinya nor Swahili.

Answer: The researcher also learns Tigrinya but not Yoruba.

Answer: The researcher also learns both Tigrinya and Yoruba.

Answer: The researcher also learns Yoruba but not Tigrinya.


Question: Each of the following could be a complete and accurate list of the researchers who learn both Swahili and Yoruba EXCEPT:

Answer: the historian

Answer: the paleontologist

Answer: the historian, the linguist

Answer: the historian, the paleontologist

Answer: the linguist, the paleontologist


Question: If the geologist learns exactly two of the languages, then which one of the following could be true?

Answer: The paleontologist learns Rundi.

Answer: The paleontologist learns Swahili.

Answer: The historian learns Rundi.

Answer: The paleontologist learns exactly three of the languages.

Answer: The historian learns exactly two of the languages.


Question: Which one of the following must be true?

Answer: Fewer of the languages are learned by the historian than are learned by the paleontologist.

Answer: Fewer of the languages are learned by the geologist than are learned by the historian.

Answer: Fewer of the languages are learned by the geologist than are learned by the linguist.

Answer: Fewer of the languages are learned by the paleontologist than are learned by the linguist.

Answer: Fewer of the languages are learned by the paleontologist than are learned by the historian.


Question: If exactly two of the languages are learned by the historian, then which one of the following must be true?

Answer: The paleontologist does not learn Rundi.

Answer: The geologist does not learn Swahili.

Answer: The linguist does not learn Rundi.

Answer: The historian does not learn Rundi.

Answer: The paleontologist does not learn Swahili.


Context: During three days—Monday through Wednesday—a health officer will inspect exactly six buildings—three hotels: Grace, Jacaranda, and Lido; and three restaurants: Seville, Vesuvio, and Zeno. Each day, exactly two buildings are inspected: one in the morning and one in the afternoon. Inspections must occur according to the following conditions: Hotels are not inspected on Wednesday. Grace is inspected at some time before Jacaranda. Grace is not inspected on the same day as Seville. If Zeno is inspected in the morning, Lido is also inspected in the morning.

Question: Which one of the following could be the order in which the buildings are inspected, listed in order from Monday morning through Wednesday afternoon?

Answer: Grace, Seville, Jacaranda, Lido, Vesuvio, Zeno

Answer: Grace, Vesuvio, Zeno, Jacaranda, Lido, Seville

Answer: Lido, Jacaranda, Grace, Vesuvio, Zeno, Seville

Answer: Lido, Seville, Grace, Jacaranda, Zeno, Vesuvio

Answer: Zeno, Grace, Jacaranda, Lido, Seville, Vesuvio


Question: Which one of the following could be the buildings inspected in the mornings, listed in order from Monday through Wednesday?

Answer: Grace, Jacaranda, Zeno

Answer: Jacaranda, Vesuvio, Seville

Answer: Lido, Jacaranda, Vesuvio

Answer: Seville, Jacaranda, Vesuvio

Answer: Seville, Lido, Zeno


Question: Which one of the following is a pair of buildings that, if inspected on the same day as each other, must be inspected on Monday?

Answer: Grace and Jacaranda

Answer: Grace and Vesuvio

Answer: Jacaranda and Lido

Answer: Lido and Seville

Answer: Lido and Vesuvio


Question: If Grace is inspected on Tuesday, which one of the following could be the buildings inspected in the afternoons, listed in order from Monday through Wednesday?

Answer: Lido, Jacaranda, Vesuvio

Answer: Lido, Jacaranda, Zeno

Answer: Lido, Vesuvio, Zeno

Answer: Seville, Grace, Vesuvio

Answer: Seville, Jacaranda, Lido


Question: If Seville is inspected on Monday morning, which one of the following must be true?

Answer: Grace is inspected on Tuesday afternoon.

Answer: Jacaranda is inspected on Monday afternoon.

Answer: Lido is inspected on Tuesday morning.

Answer: Vesuvio is inspected on Wednesday morning.

Answer: Zeno is inspected on Wednesday morning.


Question: If Grace is inspected on Monday morning and Zeno is inspected on Wednesday morning, which one of the following must be true?

Answer: Jacaranda is inspected before Lido is inspected.

Answer: Jacaranda is inspected after Lido is inspected.

Answer: Jacaranda is inspected after Seville is inspected.

Answer: Lido is inspected before Seville is inspected.

Answer: Lido is inspected before Vesuvio is inspected.


Context: Morrisville's town council has exactly three members: Fu, Gianola, and Herstein. During one week, the council members vote on exactly three bills: a recreation bill, a school bill, and a tax bill. Each council member votes either for or against each bill. The following is known: Each member of the council votes for at least one of the bills and against at least one of the bills. Exactly two members of the council vote for the recreation bill. Exactly one member of the council votes for the school bill. Exactly one member of the council votes for the tax bill. Fu votes for the recreation bill and against the school bill. Gianola votes against the recreation bill. Herstein votes against the tax bill.

Question: Which one of the following statements could be true?

Answer: Fu and Gianola vote the same way on the tax bill.

Answer: Gianola and Herstein vote the same way on the recreation bill.

Answer: Gianola and Herstein vote the same way on the school bill.

Answer: Fu votes for one of the bills and Gianola votes for two of the bills.

Answer: Fu votes for two of the bills and Gianola votes for two of the bills.


Question: If the set of members of the council who vote against the school bill is the same set of members who vote against the tax bill, then which one of the following statements must be true?

Answer: Fu votes for the tax bill.

Answer: Gianola votes for the recreation bill.

Answer: Gianola votes against the school bill.

Answer: Herstein votes against the recreation bill.

Answer: Herstein votes against the school bill.


Question: If Gianola votes for the tax bill, then which one of the following statements could be true?

Answer: Fu and Gianola each vote for exactly one bill.

Answer: Gianola and Herstein each vote for exactly one bill.

Answer: Fu votes for exactly two bills.

Answer: Gianola votes for the recreation bill.

Answer: Herstein votes against the recreation bill.


Question: If Gianola votes for exactly two of the three bills, which one of the following statements must be true?

Answer: Fu votes for the tax bill.

Answer: Gianola votes for the recreation bill.

Answer: Gianola votes for the school bill.

Answer: Gianola votes against the tax bill.

Answer: Herstein votes for the school bill.


Question: If one of the members of the council votes against exactly the same bills as does another member of the council, then which one of the following statements must be true?

Answer: Fu votes for the tax bill.

Answer: Gianola votes for the recreation bill.

Answer: Gianola votes against the school bill.

Answer: Gianola votes for exactly one bill

Answer: Herstein votes for exactly one bill.


Context: On a Tuesday, an accountant has exactly seven bills—numbered 1 through 7—to pay by Thursday of the same week. The accountant will pay each bill only once according to the following rules: Either three or four of the seven bills must be paid on Wednesday, the rest on Thursday. Bill 1 cannot be paid on the same day as bill 5. Bill 2 must be paid on Thursday. Bill 4 must be paid on the same day as bill 7. If bill 6 is paid on Wednesday, bill 7 must be paid on Thursday.

Question: If exactly four bills are paid on Wednesday, then those four bills could be

Answer: 1, 3, 4, and 6

Answer: 1, 3, 5, and 6

Answer: 2, 4, 5, and 7

Answer: 3, 4, 5, and 7

Answer: 3, 4, 6, and 7


Question: Which one of the following is a complete and accurate list of the bills any one of which could be among the bills paid on Wednesday?

Answer: 3, 5, and 6

Answer: 1, 3, 4, 6, and 7

Answer: 1, 3, 4, 5, 6, and 7

Answer: 2, 3, 4, 5, 6, and 7

Answer: 1, 2, 3, 4, 5, 6, and 7


Question: If bill 2 and bill 6 are paid on different days from each other, which one of the following must be true?

Answer: Exactly three bills are paid on Wednesday.

Answer: Exactly three bills are paid on Thursday.

Answer: Bill 1 is paid on the same day as bill 4.

Answer: Bill 2 is paid on the same day as bill 3.

Answer: Bill 5 is paid on the same day as bill 7.


Question: If bill 6 is paid on Wednesday, which one of the following bills must also be paid on Wednesday?

Answer: 1

Answer: 3

Answer: 4

Answer: 5

Answer: 7


Question: If bill 4 is paid on Thursday, which one of the following is a pair of bills that could also be paid on Thursday?

Answer: 1 and 5

Answer: 1 and 7

Answer: 3 and 5

Answer: 3 and 6

Answer: 6 and 7