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John receives one grade for each of the following six courses: economics, geology, history, Italian, physics, and Russian. From highest to lowest, the possible grades are A, B, C, D, and E. E is the only failing grade. Two letter grades are consecutive if and only if they are adjacent in the alphabet. John's grades in geology and physics are consecutive. His grades in Italian and Russian are consecutive. He receives a higher grade in economics than in history. He receives a higher grade in geology than in physics.

Assume that John's grade in physics is higher than his grade in Italian and consecutive with it and that his grades in Russian and physics differ. Which one of the following must be true?

John receives one grade for each of the following six courses: economics, geology, history, Italian, physics, and Russian. From highest to lowest, the possible grades are A, B, C, D, and E. E is the only failing grade. Two letter grades are consecutive if and only if they are adjacent in the alphabet. John's grades in geology and physics are consecutive. His grades in Italian and Russian are consecutive. He receives a higher grade in economics than in history. He receives a higher grade in geology than in physics.

Assume that John receives a lower grade in economics than in physics. He must have failed at least one course if which one of the following is also true?

A store sells shirts only in small, medium, and large sizes, and only in red, yellow, and blue colors. Casey buys exactly three shirts from the store. A shirt type consists of both a size and a color. Casey does not buy two shirts of the same type. Casey does not buy both a small shirt and a large shirt. No small red shirts are available. No large blue shirts are available.

Which one of the following must be false?

A store sells shirts only in small, medium, and large sizes, and only in red, yellow, and blue colors. Casey buys exactly three shirts from the store. A shirt type consists of both a size and a color. Casey does not buy two shirts of the same type. Casey does not buy both a small shirt and a large shirt. No small red shirts are available. No large blue shirts are available.

If Casey buys a small blue shirt, which one of the following must be false?

A store sells shirts only in small, medium, and large sizes, and only in red, yellow, and blue colors. Casey buys exactly three shirts from the store. A shirt type consists of both a size and a color. Casey does not buy two shirts of the same type. Casey does not buy both a small shirt and a large shirt. No small red shirts are available. No large blue shirts are available.

If Casey does not buy a medium yellow shirt, which one of the following must be true?

A store sells shirts only in small, medium, and large sizes, and only in red, yellow, and blue colors. Casey buys exactly three shirts from the store. A shirt type consists of both a size and a color. Casey does not buy two shirts of the same type. Casey does not buy both a small shirt and a large shirt. No small red shirts are available. No large blue shirts are available.

If Casey buys exactly one medium shirt and does not buy two shirts of the same color, then she cannot buy which one of the following?

A store sells shirts only in small, medium, and large sizes, and only in red, yellow, and blue colors. Casey buys exactly three shirts from the store. A shirt type consists of both a size and a color. Casey does not buy two shirts of the same type. Casey does not buy both a small shirt and a large shirt. No small red shirts are available. No large blue shirts are available.

If neither large red shirts nor small blue shirts are available, which one of the following must Casey buy?

A hobbyist is stocking her aquarium with exactly three fish of different types and with exactly two species of plants. The only fish under consideration are a G, an H, a J, a K, and an L, and the only kinds of plants under consideration are of the species W, X, Y, and Z. She will observe the following conditions: If she selects the G, she can select neither the H nor a Y. She cannot select the H unless she selects the K. She cannot select the J unless she selects a W. If she selects the K, she must select an X.

Which one of the following is an acceptable selection of fish and plants for the aquarium? Fish Plants

A hobbyist is stocking her aquarium with exactly three fish of different types and with exactly two species of plants. The only fish under consideration are a G, an H, a J, a K, and an L, and the only kinds of plants under consideration are of the species W, X, Y, and Z. She will observe the following conditions: If she selects the G, she can select neither the H nor a Y. She cannot select the H unless she selects the K. She cannot select the J unless she selects a W. If she selects the K, she must select an X.

If the hobbyist selects the H, which one of the following must also be true?

A hobbyist is stocking her aquarium with exactly three fish of different types and with exactly two species of plants. The only fish under consideration are a G, an H, a J, a K, and an L, and the only kinds of plants under consideration are of the species W, X, Y, and Z. She will observe the following conditions: If she selects the G, she can select neither the H nor a Y. She cannot select the H unless she selects the K. She cannot select the J unless she selects a W. If she selects the K, she must select an X.

If the hobbyist selects both X's and Z's, which one of the following could be the group of fish she selects?

A hobbyist is stocking her aquarium with exactly three fish of different types and with exactly two species of plants. The only fish under consideration are a G, an H, a J, a K, and an L, and the only kinds of plants under consideration are of the species W, X, Y, and Z. She will observe the following conditions: If she selects the G, she can select neither the H nor a Y. She cannot select the H unless she selects the K. She cannot select the J unless she selects a W. If she selects the K, she must select an X.

The hobbyist could select any of the following groups of fish for the aquarium EXCEPT

A hobbyist is stocking her aquarium with exactly three fish of different types and with exactly two species of plants. The only fish under consideration are a G, an H, a J, a K, and an L, and the only kinds of plants under consideration are of the species W, X, Y, and Z. She will observe the following conditions: If she selects the G, she can select neither the H nor a Y. She cannot select the H unless she selects the K. She cannot select the J unless she selects a W. If she selects the K, she must select an X.

If the hobbyist selects a Y, which one of the following must be the group of fish she selects?

A hobbyist is stocking her aquarium with exactly three fish of different types and with exactly two species of plants. The only fish under consideration are a G, an H, a J, a K, and an L, and the only kinds of plants under consideration are of the species W, X, Y, and Z. She will observe the following conditions: If she selects the G, she can select neither the H nor a Y. She cannot select the H unless she selects the K. She cannot select the J unless she selects a W. If she selects the K, she must select an X.

The hobbyist could select any of the following plant combinations EXCEPT

A committee ranks five towns—Palmdale, Quietville, Riverdale, Seaside, Tidetown—from first (best) to fifth (worst) on each of three criteria: climate, location, friendliness. For each of the three criteria, none of the five towns receives the same ranking as any other town does. In climate, Tidetown is ranked third, and Seaside fourth. In location, Quietville is ranked second, Riverdale third, Palmdale fourth. In friendliness, Tidetown's ranking is better than Palmdale's, Quietville is ranked fourth and Seaside fifth. Riverdale receives a better ranking in climate than in friendliness. Quietville's three ranking are all different from each other.

Which of the following is a complete and accurate list of the rankings any one of which could be the ranking on climate given to Riverdale?

A committee ranks five towns—Palmdale, Quietville, Riverdale, Seaside, Tidetown—from first (best) to fifth (worst) on each of three criteria: climate, location, friendliness. For each of the three criteria, none of the five towns receives the same ranking as any other town does. In climate, Tidetown is ranked third, and Seaside fourth. In location, Quietville is ranked second, Riverdale third, Palmdale fourth. In friendliness, Tidetown's ranking is better than Palmdale's, Quietville is ranked fourth and Seaside fifth. Riverdale receives a better ranking in climate than in friendliness. Quietville's three ranking are all different from each other.

Which of the following is a town that CANNOT be ranked fifth on any one of the three criteria?

A committee ranks five towns—Palmdale, Quietville, Riverdale, Seaside, Tidetown—from first (best) to fifth (worst) on each of three criteria: climate, location, friendliness. For each of the three criteria, none of the five towns receives the same ranking as any other town does. In climate, Tidetown is ranked third, and Seaside fourth. In location, Quietville is ranked second, Riverdale third, Palmdale fourth. In friendliness, Tidetown's ranking is better than Palmdale's, Quietville is ranked fourth and Seaside fifth. Riverdale receives a better ranking in climate than in friendliness. Quietville's three ranking are all different from each other.

Which of the following could be true?

A committee ranks five towns—Palmdale, Quietville, Riverdale, Seaside, Tidetown—from first (best) to fifth (worst) on each of three criteria: climate, location, friendliness. For each of the three criteria, none of the five towns receives the same ranking as any other town does. In climate, Tidetown is ranked third, and Seaside fourth. In location, Quietville is ranked second, Riverdale third, Palmdale fourth. In friendliness, Tidetown's ranking is better than Palmdale's, Quietville is ranked fourth and Seaside fifth. Riverdale receives a better ranking in climate than in friendliness. Quietville's three ranking are all different from each other.

If Quietville is ranked first in climate, then it must be true that

A committee ranks five towns—Palmdale, Quietville, Riverdale, Seaside, Tidetown—from first (best) to fifth (worst) on each of three criteria: climate, location, friendliness. For each of the three criteria, none of the five towns receives the same ranking as any other town does. In climate, Tidetown is ranked third, and Seaside fourth. In location, Quietville is ranked second, Riverdale third, Palmdale fourth. In friendliness, Tidetown's ranking is better than Palmdale's, Quietville is ranked fourth and Seaside fifth. Riverdale receives a better ranking in climate than in friendliness. Quietville's three ranking are all different from each other.

If Palmdale is ranked second in climate, then which one of the following can be true?

A committee ranks five towns—Palmdale, Quietville, Riverdale, Seaside, Tidetown—from first (best) to fifth (worst) on each of three criteria: climate, location, friendliness. For each of the three criteria, none of the five towns receives the same ranking as any other town does. In climate, Tidetown is ranked third, and Seaside fourth. In location, Quietville is ranked second, Riverdale third, Palmdale fourth. In friendliness, Tidetown's ranking is better than Palmdale's, Quietville is ranked fourth and Seaside fifth. Riverdale receives a better ranking in climate than in friendliness. Quietville's three ranking are all different from each other.

If Tidetown is ranked first in location and Riverdale is ranked second in friendliness, then it is possible to deduce with certainty all three rankings for exactly how many of the towns?

A committee ranks five towns—Palmdale, Quietville, Riverdale, Seaside, Tidetown—from first (best) to fifth (worst) on each of three criteria: climate, location, friendliness. For each of the three criteria, none of the five towns receives the same ranking as any other town does. In climate, Tidetown is ranked third, and Seaside fourth. In location, Quietville is ranked second, Riverdale third, Palmdale fourth. In friendliness, Tidetown's ranking is better than Palmdale's, Quietville is ranked fourth and Seaside fifth. Riverdale receives a better ranking in climate than in friendliness. Quietville's three ranking are all different from each other.

Which one of the following statements CANNOT be true?

Petworld has exactly fourteen animals (three gerbils, three hamsters, three lizards, five snakes) that are kept in four separate cages (W, X, Y, Z) according to the following conditions: Each cage contains exactly two, four, or six animals. Any cage containing a gerbil also contains at least one hamster; any cage containing a hamster also contains at least one gerbil. Any cage containing a lizard also contains at least one snake; any cage containing a snake also contains at least one lizard. Neither cage Y nor cage Z contains a gerbil. Neither cage W nor cage X contains a lizard.

Which one of the following could be a complete and accurate list of the animals kept in cages W and Y?

Petworld has exactly fourteen animals (three gerbils, three hamsters, three lizards, five snakes) that are kept in four separate cages (W, X, Y, Z) according to the following conditions: Each cage contains exactly two, four, or six animals. Any cage containing a gerbil also contains at least one hamster; any cage containing a hamster also contains at least one gerbil. Any cage containing a lizard also contains at least one snake; any cage containing a snake also contains at least one lizard. Neither cage Y nor cage Z contains a gerbil. Neither cage W nor cage X contains a lizard.

If there are exactly two hamsters in cage W and the number of gerbils in cage X is equal to the number of snakes in cage Y, then the number of snakes in cage Z must be exactly

Petworld has exactly fourteen animals (three gerbils, three hamsters, three lizards, five snakes) that are kept in four separate cages (W, X, Y, Z) according to the following conditions: Each cage contains exactly two, four, or six animals. Any cage containing a gerbil also contains at least one hamster; any cage containing a hamster also contains at least one gerbil. Any cage containing a lizard also contains at least one snake; any cage containing a snake also contains at least one lizard. Neither cage Y nor cage Z contains a gerbil. Neither cage W nor cage X contains a lizard.

If cage Z contains exactly twice as many lizards as cage Y, which one of the following can be true?

Petworld has exactly fourteen animals (three gerbils, three hamsters, three lizards, five snakes) that are kept in four separate cages (W, X, Y, Z) according to the following conditions: Each cage contains exactly two, four, or six animals. Any cage containing a gerbil also contains at least one hamster; any cage containing a hamster also contains at least one gerbil. Any cage containing a lizard also contains at least one snake; any cage containing a snake also contains at least one lizard. Neither cage Y nor cage Z contains a gerbil. Neither cage W nor cage X contains a lizard.

If the number of animals in cage W is equal to the number of animals in cage Z, then which one of the following can be true?

Petworld has exactly fourteen animals (three gerbils, three hamsters, three lizards, five snakes) that are kept in four separate cages (W, X, Y, Z) according to the following conditions: Each cage contains exactly two, four, or six animals. Any cage containing a gerbil also contains at least one hamster; any cage containing a hamster also contains at least one gerbil. Any cage containing a lizard also contains at least one snake; any cage containing a snake also contains at least one lizard. Neither cage Y nor cage Z contains a gerbil. Neither cage W nor cage X contains a lizard.

If cage Y contains six animals, which one of the following must be true?

Petworld has exactly fourteen animals (three gerbils, three hamsters, three lizards, five snakes) that are kept in four separate cages (W, X, Y, Z) according to the following conditions: Each cage contains exactly two, four, or six animals. Any cage containing a gerbil also contains at least one hamster; any cage containing a hamster also contains at least one gerbil. Any cage containing a lizard also contains at least one snake; any cage containing a snake also contains at least one lizard. Neither cage Y nor cage Z contains a gerbil. Neither cage W nor cage X contains a lizard.

At most, how many snakes can occupy cage Y at any one time?

A soft drink manufacturer surveyed consumer preferences for exactly seven proposed names for its new soda: Jazz, Kola, Luck, Mist, Nipi, Oboy, and Ping. The manufacturer ranked the seven names according to the number of votes they received. The name that received the most votes was ranked first. Every name received a different number of votes. Some of the survey results are as follows: Jazz received more votes than Oboy. Oboy received more votes than Kola. Kola received more votes than Mist. Nipi did not receive the fewest votes. Ping received fewer votes than Luck but more votes than Nipi and more votes than Oboy.

Which one of the following could be an accurate list of the seven names in rank order from first through seventh?

A soft drink manufacturer surveyed consumer preferences for exactly seven proposed names for its new soda: Jazz, Kola, Luck, Mist, Nipi, Oboy, and Ping. The manufacturer ranked the seven names according to the number of votes they received. The name that received the most votes was ranked first. Every name received a different number of votes. Some of the survey results are as follows: Jazz received more votes than Oboy. Oboy received more votes than Kola. Kola received more votes than Mist. Nipi did not receive the fewest votes. Ping received fewer votes than Luck but more votes than Nipi and more votes than Oboy.

Which one of the following statements must be true?

A soft drink manufacturer surveyed consumer preferences for exactly seven proposed names for its new soda: Jazz, Kola, Luck, Mist, Nipi, Oboy, and Ping. The manufacturer ranked the seven names according to the number of votes they received. The name that received the most votes was ranked first. Every name received a different number of votes. Some of the survey results are as follows: Jazz received more votes than Oboy. Oboy received more votes than Kola. Kola received more votes than Mist. Nipi did not receive the fewest votes. Ping received fewer votes than Luck but more votes than Nipi and more votes than Oboy.

If the ranks of Ping, Oboy, and Kola were consecutive, then which one of the following statements would have to be false?

A soft drink manufacturer surveyed consumer preferences for exactly seven proposed names for its new soda: Jazz, Kola, Luck, Mist, Nipi, Oboy, and Ping. The manufacturer ranked the seven names according to the number of votes they received. The name that received the most votes was ranked first. Every name received a different number of votes. Some of the survey results are as follows: Jazz received more votes than Oboy. Oboy received more votes than Kola. Kola received more votes than Mist. Nipi did not receive the fewest votes. Ping received fewer votes than Luck but more votes than Nipi and more votes than Oboy.

What is the total number of the soft drink names whose exact ranks can be deduced from the partial survey results?

A soft drink manufacturer surveyed consumer preferences for exactly seven proposed names for its new soda: Jazz, Kola, Luck, Mist, Nipi, Oboy, and Ping. The manufacturer ranked the seven names according to the number of votes they received. The name that received the most votes was ranked first. Every name received a different number of votes. Some of the survey results are as follows: Jazz received more votes than Oboy. Oboy received more votes than Kola. Kola received more votes than Mist. Nipi did not receive the fewest votes. Ping received fewer votes than Luck but more votes than Nipi and more votes than Oboy.

What is the maximum possible number of the soft drink names any one of which could be among the three most popular?

A soft drink manufacturer surveyed consumer preferences for exactly seven proposed names for its new soda: Jazz, Kola, Luck, Mist, Nipi, Oboy, and Ping. The manufacturer ranked the seven names according to the number of votes they received. The name that received the most votes was ranked first. Every name received a different number of votes. Some of the survey results are as follows: Jazz received more votes than Oboy. Oboy received more votes than Kola. Kola received more votes than Mist. Nipi did not receive the fewest votes. Ping received fewer votes than Luck but more votes than Nipi and more votes than Oboy.

If Ping received more votes than Jazz, then what is the maximum possible number of names whose ranks can be determined?

Eight benches—J, K, L, T, U, X, Y, and Z are arranged along the perimeter of a park as shown below: The following is true: J, K, and L are green; T and U are red; X, Y, and Z are pink. The green benches stand next to one another along the park's perimeter. The pink benches stand next to one another along the park's perimeter. No green bench stands next to a pink bench. The bench on the southeast corner is T. J stands at the center of the park's north side. If T stands next to X, then T does not also stand next to L.

Which one of the following benches could be on the northeast corner of the park?

Eight benches—J, K, L, T, U, X, Y, and Z are arranged along the perimeter of a park as shown below: The following is true: J, K, and L are green; T and U are red; X, Y, and Z are pink. The green benches stand next to one another along the park's perimeter. The pink benches stand next to one another along the park's perimeter. No green bench stands next to a pink bench. The bench on the southeast corner is T. J stands at the center of the park's north side. If T stands next to X, then T does not also stand next to L.

Each of the following statements must be true EXCEPT:

Eight benches—J, K, L, T, U, X, Y, and Z are arranged along the perimeter of a park as shown below: The following is true: J, K, and L are green; T and U are red; X, Y, and Z are pink. The green benches stand next to one another along the park's perimeter. The pink benches stand next to one another along the park's perimeter. No green bench stands next to a pink bench. The bench on the southeast corner is T. J stands at the center of the park's north side. If T stands next to X, then T does not also stand next to L.

Which one of the following benches must be next to J?

Eight benches—J, K, L, T, U, X, Y, and Z are arranged along the perimeter of a park as shown below: The following is true: J, K, and L are green; T and U are red; X, Y, and Z are pink. The green benches stand next to one another along the park's perimeter. The pink benches stand next to one another along the park's perimeter. No green bench stands next to a pink bench. The bench on the southeast corner is T. J stands at the center of the park's north side. If T stands next to X, then T does not also stand next to L.

For which one of the following benches are there two and no more than two locations either one of which could be the location the bench occupies?

Eight benches—J, K, L, T, U, X, Y, and Z are arranged along the perimeter of a park as shown below: The following is true: J, K, and L are green; T and U are red; X, Y, and Z are pink. The green benches stand next to one another along the park's perimeter. The pink benches stand next to one another along the park's perimeter. No green bench stands next to a pink bench. The bench on the southeast corner is T. J stands at the center of the park's north side. If T stands next to X, then T does not also stand next to L.

If Z is directly north of Y, which one of the following statements must be true?

Eight benches—J, K, L, T, U, X, Y, and Z are arranged along the perimeter of a park as shown below: The following is true: J, K, and L are green; T and U are red; X, Y, and Z are pink. The green benches stand next to one another along the park's perimeter. The pink benches stand next to one another along the park's perimeter. No green bench stands next to a pink bench. The bench on the southeast corner is T. J stands at the center of the park's north side. If T stands next to X, then T does not also stand next to L.

If Y is in the middle of the west side of the park, then the two benches in which one of the following pairs CANNOT be two of the corner benches?

Eight benches—J, K, L, T, U, X, Y, and Z are arranged along the perimeter of a park as shown below: The following is true: J, K, and L are green; T and U are red; X, Y, and Z are pink. The green benches stand next to one another along the park's perimeter. The pink benches stand next to one another along the park's perimeter. No green bench stands next to a pink bench. The bench on the southeast corner is T. J stands at the center of the park's north side. If T stands next to X, then T does not also stand next to L.

If Y is farther south than L and farther north than T, then the benches in each of the following pairs must be next to each other EXCEPT

A lake contains exactly five islands—J, K, L, M, O—which are unconnected by bridges. Contractors will build a network of bridges that satisfies the following specifications: Each bridge directly connects exactly two islands with each other, and no two bridges intersect. No more than one bridge directly connects any two islands. No island has more than three bridges that directly connect it with other islands. J, K, and L are each directly connected by bridge with one or both of M and O. J is directly connected by bridge with exactly two islands. K is directly connected by bridge with exactly one island. A bridge directly connects J with O, and a bridge directly connects M with O.

Which one of the following is a complete and accurate list of the islands any one of which could be directly connected by bridge with L ?

A lake contains exactly five islands—J, K, L, M, O—which are unconnected by bridges. Contractors will build a network of bridges that satisfies the following specifications: Each bridge directly connects exactly two islands with each other, and no two bridges intersect. No more than one bridge directly connects any two islands. No island has more than three bridges that directly connect it with other islands. J, K, and L are each directly connected by bridge with one or both of M and O. J is directly connected by bridge with exactly two islands. K is directly connected by bridge with exactly one island. A bridge directly connects J with O, and a bridge directly connects M with O.

Which one of the following could be true about the completed network of bridges?

A lake contains exactly five islands—J, K, L, M, O—which are unconnected by bridges. Contractors will build a network of bridges that satisfies the following specifications: Each bridge directly connects exactly two islands with each other, and no two bridges intersect. No more than one bridge directly connects any two islands. No island has more than three bridges that directly connect it with other islands. J, K, and L are each directly connected by bridge with one or both of M and O. J is directly connected by bridge with exactly two islands. K is directly connected by bridge with exactly one island. A bridge directly connects J with O, and a bridge directly connects M with O.

If a bridge directly connects K with O, then which one of the following could be true?

A lake contains exactly five islands—J, K, L, M, O—which are unconnected by bridges. Contractors will build a network of bridges that satisfies the following specifications: Each bridge directly connects exactly two islands with each other, and no two bridges intersect. No more than one bridge directly connects any two islands. No island has more than three bridges that directly connect it with other islands. J, K, and L are each directly connected by bridge with one or both of M and O. J is directly connected by bridge with exactly two islands. K is directly connected by bridge with exactly one island. A bridge directly connects J with O, and a bridge directly connects M with O.

If a bridge directly connects L with M and a bridge directly connects L with O, then which one of the following must be true?

A lake contains exactly five islands—J, K, L, M, O—which are unconnected by bridges. Contractors will build a network of bridges that satisfies the following specifications: Each bridge directly connects exactly two islands with each other, and no two bridges intersect. No more than one bridge directly connects any two islands. No island has more than three bridges that directly connect it with other islands. J, K, and L are each directly connected by bridge with one or both of M and O. J is directly connected by bridge with exactly two islands. K is directly connected by bridge with exactly one island. A bridge directly connects J with O, and a bridge directly connects M with O.

If no island that is directly connected by bridge with M is also directly connected by bridge with O, then there must be a bridge directly connecting

Each of five students—Hubert, Lori, Paul, Regina, and Sharon—will visit exactly one of three cities—Montreal, Toronto, or Vancouver—for the month of March, according to the following conditions: Sharon visits a different city than Paul. Hubert visits the same city as Regina. Lori visits Montreal or else Toronto. If Paul visits Vancouver, Hubert visits Vancouver with him. Each student visits one of the cities with at least one of the other four students.

Which one of the following could be true for March?

Each of five students—Hubert, Lori, Paul, Regina, and Sharon—will visit exactly one of three cities—Montreal, Toronto, or Vancouver—for the month of March, according to the following conditions: Sharon visits a different city than Paul. Hubert visits the same city as Regina. Lori visits Montreal or else Toronto. If Paul visits Vancouver, Hubert visits Vancouver with him. Each student visits one of the cities with at least one of the other four students.

If Hubert and Sharon visit a city together, which one of the following could be true in March?

Each of five students—Hubert, Lori, Paul, Regina, and Sharon—will visit exactly one of three cities—Montreal, Toronto, or Vancouver—for the month of March, according to the following conditions: Sharon visits a different city than Paul. Hubert visits the same city as Regina. Lori visits Montreal or else Toronto. If Paul visits Vancouver, Hubert visits Vancouver with him. Each student visits one of the cities with at least one of the other four students.

If Sharon visits Vancouver, which one of the following must be true for March?

Each of five students—Hubert, Lori, Paul, Regina, and Sharon—will visit exactly one of three cities—Montreal, Toronto, or Vancouver—for the month of March, according to the following conditions: Sharon visits a different city than Paul. Hubert visits the same city as Regina. Lori visits Montreal or else Toronto. If Paul visits Vancouver, Hubert visits Vancouver with him. Each student visits one of the cities with at least one of the other four students.

Which one of the following could be false in March?

Each of five students—Hubert, Lori, Paul, Regina, and Sharon—will visit exactly one of three cities—Montreal, Toronto, or Vancouver—for the month of March, according to the following conditions: Sharon visits a different city than Paul. Hubert visits the same city as Regina. Lori visits Montreal or else Toronto. If Paul visits Vancouver, Hubert visits Vancouver with him. Each student visits one of the cities with at least one of the other four students.

If Regina visits Toronto, which one of the following could be true in March?

Each of five students—Hubert, Lori, Paul, Regina, and Sharon—will visit exactly one of three cities—Montreal, Toronto, or Vancouver—for the month of March, according to the following conditions: Sharon visits a different city than Paul. Hubert visits the same city as Regina. Lori visits Montreal or else Toronto. If Paul visits Vancouver, Hubert visits Vancouver with him. Each student visits one of the cities with at least one of the other four students.

Which one of the following must be true for March?

A college offers one course in each of three subjects—mathematics, nutrition, and oceanography—in the fall and again in the spring. Students' book orders for these course offerings are kept in six folders, numbered 1 through 6, from which labels identifying the folders' contents are missing. The following is known: Each folder contains only the orders for one of the six course offerings. Folder 1 contains orders for the same subject as folder 2 does. The orders in folder 3 are for a different subject than are the orders in folder 4. The fall mathematics orders are in folder 1 or else folder 4. The spring oceanography orders are in folder 1 or else folder 4. The spring nutrition orders are not in folder 5.

Which one of the following could be the list of the contents of the folders, in order from folder 1 to folder 6 ?

A college offers one course in each of three subjects—mathematics, nutrition, and oceanography—in the fall and again in the spring. Students' book orders for these course offerings are kept in six folders, numbered 1 through 6, from which labels identifying the folders' contents are missing. The following is known: Each folder contains only the orders for one of the six course offerings. Folder 1 contains orders for the same subject as folder 2 does. The orders in folder 3 are for a different subject than are the orders in folder 4. The fall mathematics orders are in folder 1 or else folder 4. The spring oceanography orders are in folder 1 or else folder 4. The spring nutrition orders are not in folder 5.

Which one of the following statements must be false?

A college offers one course in each of three subjects—mathematics, nutrition, and oceanography—in the fall and again in the spring. Students' book orders for these course offerings are kept in six folders, numbered 1 through 6, from which labels identifying the folders' contents are missing. The following is known: Each folder contains only the orders for one of the six course offerings. Folder 1 contains orders for the same subject as folder 2 does. The orders in folder 3 are for a different subject than are the orders in folder 4. The fall mathematics orders are in folder 1 or else folder 4. The spring oceanography orders are in folder 1 or else folder 4. The spring nutrition orders are not in folder 5.

If the fall oceanography orders are in folder 2, then which one of the following statements could be true?

A college offers one course in each of three subjects—mathematics, nutrition, and oceanography—in the fall and again in the spring. Students' book orders for these course offerings are kept in six folders, numbered 1 through 6, from which labels identifying the folders' contents are missing. The following is known: Each folder contains only the orders for one of the six course offerings. Folder 1 contains orders for the same subject as folder 2 does. The orders in folder 3 are for a different subject than are the orders in folder 4. The fall mathematics orders are in folder 1 or else folder 4. The spring oceanography orders are in folder 1 or else folder 4. The spring nutrition orders are not in folder 5.

Which one of the following statements could be true?

A college offers one course in each of three subjects—mathematics, nutrition, and oceanography—in the fall and again in the spring. Students' book orders for these course offerings are kept in six folders, numbered 1 through 6, from which labels identifying the folders' contents are missing. The following is known: Each folder contains only the orders for one of the six course offerings. Folder 1 contains orders for the same subject as folder 2 does. The orders in folder 3 are for a different subject than are the orders in folder 4. The fall mathematics orders are in folder 1 or else folder 4. The spring oceanography orders are in folder 1 or else folder 4. The spring nutrition orders are not in folder 5.

If the fall oceanography orders are in folder 2, then for exactly how many of the remaining five folders can it be deduced which course offering's orders are in that folder?

A college offers one course in each of three subjects—mathematics, nutrition, and oceanography—in the fall and again in the spring. Students' book orders for these course offerings are kept in six folders, numbered 1 through 6, from which labels identifying the folders' contents are missing. The following is known: Each folder contains only the orders for one of the six course offerings. Folder 1 contains orders for the same subject as folder 2 does. The orders in folder 3 are for a different subject than are the orders in folder 4. The fall mathematics orders are in folder 1 or else folder 4. The spring oceanography orders are in folder 1 or else folder 4. The spring nutrition orders are not in folder 5.

Which one of the following lists a pair of folders that must together contain orders for two different subjects?

A college offers one course in each of three subjects—mathematics, nutrition, and oceanography—in the fall and again in the spring. Students' book orders for these course offerings are kept in six folders, numbered 1 through 6, from which labels identifying the folders' contents are missing. The following is known: Each folder contains only the orders for one of the six course offerings. Folder 1 contains orders for the same subject as folder 2 does. The orders in folder 3 are for a different subject than are the orders in folder 4. The fall mathematics orders are in folder 1 or else folder 4. The spring oceanography orders are in folder 1 or else folder 4. The spring nutrition orders are not in folder 5.

Which one of the following could be true?

Greenburg has exactly five subway lines: Ll, L2, L3, L4, and L5. Along each of the lines, trains run in both directions, stopping at every station. Ll runs in a loop connecting exactly seven stations, their order being Rincon-Tonka-French-Semplain-Urstine-Quetzal-Park-Rincon in one direction of travel, and the reverse in the other direction. L2 connects Tonka with Semplain, and with no other station. L3 connects Rincon with Urstine, and with no other station. L4 runs from Quetzal through exactly one other station, Greene, to Rincon. L5 connects Quetzal with Tonka, and with no other station.

How many different stations are there that a traveler starting at Rincon could reach by using the subway lines without making any intermediate stops?

Greenburg has exactly five subway lines: Ll, L2, L3, L4, and L5. Along each of the lines, trains run in both directions, stopping at every station. Ll runs in a loop connecting exactly seven stations, their order being Rincon-Tonka-French-Semplain-Urstine-Quetzal-Park-Rincon in one direction of travel, and the reverse in the other direction. L2 connects Tonka with Semplain, and with no other station. L3 connects Rincon with Urstine, and with no other station. L4 runs from Quetzal through exactly one other station, Greene, to Rincon. L5 connects Quetzal with Tonka, and with no other station.

In order to go from Greene to Semplain taking the fewest possible subway lines and making the fewest possible stops, a traveler must make a stop at

Greenburg has exactly five subway lines: Ll, L2, L3, L4, and L5. Along each of the lines, trains run in both directions, stopping at every station. Ll runs in a loop connecting exactly seven stations, their order being Rincon-Tonka-French-Semplain-Urstine-Quetzal-Park-Rincon in one direction of travel, and the reverse in the other direction. L2 connects Tonka with Semplain, and with no other station. L3 connects Rincon with Urstine, and with no other station. L4 runs from Quetzal through exactly one other station, Greene, to Rincon. L5 connects Quetzal with Tonka, and with no other station.

If L3 is not running and a traveler goes by subway from Urstine to Rincon making the fewest possible stops, which one of the following lists all of the intermediate stations in sequence along one of the routes that the traveler could take?

Greenburg has exactly five subway lines: Ll, L2, L3, L4, and L5. Along each of the lines, trains run in both directions, stopping at every station. Ll runs in a loop connecting exactly seven stations, their order being Rincon-Tonka-French-Semplain-Urstine-Quetzal-Park-Rincon in one direction of travel, and the reverse in the other direction. L2 connects Tonka with Semplain, and with no other station. L3 connects Rincon with Urstine, and with no other station. L4 runs from Quetzal through exactly one other station, Greene, to Rincon. L5 connects Quetzal with Tonka, and with no other station.

In order to go by subway from French to Greene, the minimum number of intermediate stops a traveler must make is

Greenburg has exactly five subway lines: Ll, L2, L3, L4, and L5. Along each of the lines, trains run in both directions, stopping at every station. Ll runs in a loop connecting exactly seven stations, their order being Rincon-Tonka-French-Semplain-Urstine-Quetzal-Park-Rincon in one direction of travel, and the reverse in the other direction. L2 connects Tonka with Semplain, and with no other station. L3 connects Rincon with Urstine, and with no other station. L4 runs from Quetzal through exactly one other station, Greene, to Rincon. L5 connects Quetzal with Tonka, and with no other station.

If the tracks that directly connect Urstine and Quetzal are blocked in both directions, a traveler going from Semplain to Park and making the fewest possible intermediate stops must pass through

Greenburg has exactly five subway lines: Ll, L2, L3, L4, and L5. Along each of the lines, trains run in both directions, stopping at every station. Ll runs in a loop connecting exactly seven stations, their order being Rincon-Tonka-French-Semplain-Urstine-Quetzal-Park-Rincon in one direction of travel, and the reverse in the other direction. L2 connects Tonka with Semplain, and with no other station. L3 connects Rincon with Urstine, and with no other station. L4 runs from Quetzal through exactly one other station, Greene, to Rincon. L5 connects Quetzal with Tonka, and with no other station.

If a sixth subway line is to be constructed so that all of the stations would have two or more lines reaching them, the stations connected by the new subway line must include at least

Prior to this year's annual promotion review, the staff of a law firm consisted of partners Harrison and Rafael, associate Olivos, and assistants Ganz, Johnson, Lowry, Stefano, Turner, and Wilford. During each annual review, each assistant and associate is considered for promotion to the next higher rank, and at least one person is promoted from each of the two lower ranks. An assistant is promoted to associate when a majority of higherranking staff votes for promotion. An associate is promoted to partner when a majority of partners vote for promotion. Everyone eligible votes on every promotion. No one joins or leaves the firm. Olivos never votes for promoting Ganz, Johnson, or Turner. Rafael never votes for promoting Lowry or Stefano. Harrison never votes for promoting Johnson or Wilford.

Which one of the following could be the distribution of staff resulting from this year's review?

Prior to this year's annual promotion review, the staff of a law firm consisted of partners Harrison and Rafael, associate Olivos, and assistants Ganz, Johnson, Lowry, Stefano, Turner, and Wilford. During each annual review, each assistant and associate is considered for promotion to the next higher rank, and at least one person is promoted from each of the two lower ranks. An assistant is promoted to associate when a majority of higherranking staff votes for promotion. An associate is promoted to partner when a majority of partners vote for promotion. Everyone eligible votes on every promotion. No one joins or leaves the firm. Olivos never votes for promoting Ganz, Johnson, or Turner. Rafael never votes for promoting Lowry or Stefano. Harrison never votes for promoting Johnson or Wilford.

If Rafael votes for promoting only Ganz, Olivos, and Wilford, and if Harrison votes for promoting only Lowry, Olivos, and Stefano, then which one of the following could be the complete roster of associates resulting from this year's review?

Prior to this year's annual promotion review, the staff of a law firm consisted of partners Harrison and Rafael, associate Olivos, and assistants Ganz, Johnson, Lowry, Stefano, Turner, and Wilford. During each annual review, each assistant and associate is considered for promotion to the next higher rank, and at least one person is promoted from each of the two lower ranks. An assistant is promoted to associate when a majority of higherranking staff votes for promotion. An associate is promoted to partner when a majority of partners vote for promotion. Everyone eligible votes on every promotion. No one joins or leaves the firm. Olivos never votes for promoting Ganz, Johnson, or Turner. Rafael never votes for promoting Lowry or Stefano. Harrison never votes for promoting Johnson or Wilford.

If Johnson is to be promoted to associate during next year's review, which one of the following is the smallest number of assistants who must be promoted during this year's review?

Prior to this year's annual promotion review, the staff of a law firm consisted of partners Harrison and Rafael, associate Olivos, and assistants Ganz, Johnson, Lowry, Stefano, Turner, and Wilford. During each annual review, each assistant and associate is considered for promotion to the next higher rank, and at least one person is promoted from each of the two lower ranks. An assistant is promoted to associate when a majority of higherranking staff votes for promotion. An associate is promoted to partner when a majority of partners vote for promotion. Everyone eligible votes on every promotion. No one joins or leaves the firm. Olivos never votes for promoting Ganz, Johnson, or Turner. Rafael never votes for promoting Lowry or Stefano. Harrison never votes for promoting Johnson or Wilford.

Which one of the following must be true after next year's review?

Prior to this year's annual promotion review, the staff of a law firm consisted of partners Harrison and Rafael, associate Olivos, and assistants Ganz, Johnson, Lowry, Stefano, Turner, and Wilford. During each annual review, each assistant and associate is considered for promotion to the next higher rank, and at least one person is promoted from each of the two lower ranks. An assistant is promoted to associate when a majority of higherranking staff votes for promotion. An associate is promoted to partner when a majority of partners vote for promotion. Everyone eligible votes on every promotion. No one joins or leaves the firm. Olivos never votes for promoting Ganz, Johnson, or Turner. Rafael never votes for promoting Lowry or Stefano. Harrison never votes for promoting Johnson or Wilford.

What is the smallest possible number of associates in the firm immediately after next year's review?

Seven consecutive time slots for a broadcast, numbered in chronological order 1 through 7, will be filled by six song tapes—G, H, L, O, P, S—and exactly one news tape. Each tape is to be assigned to a different time slot, and no tape is longer than any other tape. The broadcast is subject to the following restrictions: L must be played immediately before O. The news tape must be played at some time after L. There must be exactly two time slots between G and P, regardless of whether G comes before P or whether G comes after P.

If G is played second, which one of the following tapes must be played third?

Seven consecutive time slots for a broadcast, numbered in chronological order 1 through 7, will be filled by six song tapes—G, H, L, O, P, S—and exactly one news tape. Each tape is to be assigned to a different time slot, and no tape is longer than any other tape. The broadcast is subject to the following restrictions: L must be played immediately before O. The news tape must be played at some time after L. There must be exactly two time slots between G and P, regardless of whether G comes before P or whether G comes after P.

The news tape can be played in any one of the following time slots EXCEPT the

Seven consecutive time slots for a broadcast, numbered in chronological order 1 through 7, will be filled by six song tapes—G, H, L, O, P, S—and exactly one news tape. Each tape is to be assigned to a different time slot, and no tape is longer than any other tape. The broadcast is subject to the following restrictions: L must be played immediately before O. The news tape must be played at some time after L. There must be exactly two time slots between G and P, regardless of whether G comes before P or whether G comes after P.

If H and S are to be scheduled as far from each other as possible, then the first, the second, and the third time slots could be filled, respectively, by

Seven consecutive time slots for a broadcast, numbered in chronological order 1 through 7, will be filled by six song tapes—G, H, L, O, P, S—and exactly one news tape. Each tape is to be assigned to a different time slot, and no tape is longer than any other tape. The broadcast is subject to the following restrictions: L must be played immediately before O. The news tape must be played at some time after L. There must be exactly two time slots between G and P, regardless of whether G comes before P or whether G comes after P.

If P is played fifth, L must be played

Seven consecutive time slots for a broadcast, numbered in chronological order 1 through 7, will be filled by six song tapes—G, H, L, O, P, S—and exactly one news tape. Each tape is to be assigned to a different time slot, and no tape is longer than any other tape. The broadcast is subject to the following restrictions: L must be played immediately before O. The news tape must be played at some time after L. There must be exactly two time slots between G and P, regardless of whether G comes before P or whether G comes after P.

What is the maximum number of tapes that can separate S from the news?

Seven consecutive time slots for a broadcast, numbered in chronological order 1 through 7, will be filled by six song tapes—G, H, L, O, P, S—and exactly one news tape. Each tape is to be assigned to a different time slot, and no tape is longer than any other tape. The broadcast is subject to the following restrictions: L must be played immediately before O. The news tape must be played at some time after L. There must be exactly two time slots between G and P, regardless of whether G comes before P or whether G comes after P.

Which one of the following is the latest time slot in which L can be played?

Seven consecutive time slots for a broadcast, numbered in chronological order 1 through 7, will be filled by six song tapes—G, H, L, O, P, S—and exactly one news tape. Each tape is to be assigned to a different time slot, and no tape is longer than any other tape. The broadcast is subject to the following restrictions: L must be played immediately before O. The news tape must be played at some time after L. There must be exactly two time slots between G and P, regardless of whether G comes before P or whether G comes after P.

The time slot in which O must be played is completely determined if G is assigned to which one of the following time slots?

Doctor Yamata works only on Mondays, Tuesdays, Wednesdays, Fridays, and Saturdays. She performs four different activities—lecturing, operating, treating patients, and conducting research. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. During each week her work schedule must satisfy the following restrictions: She performs operations on exactly three mornings. If she operates on Monday, she does not operate on Tuesday. She lectures in the afternoon on exactly two consecutive calendar days. She treats patients on exactly one morning and exactly three afternoons. She conducts research on exactly one morning. On Saturday she neither lectures nor performs operations.

Which one of the following must be a day on which Doctor Yamata lectures?

Doctor Yamata works only on Mondays, Tuesdays, Wednesdays, Fridays, and Saturdays. She performs four different activities—lecturing, operating, treating patients, and conducting research. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. During each week her work schedule must satisfy the following restrictions: She performs operations on exactly three mornings. If she operates on Monday, she does not operate on Tuesday. She lectures in the afternoon on exactly two consecutive calendar days. She treats patients on exactly one morning and exactly three afternoons. She conducts research on exactly one morning. On Saturday she neither lectures nor performs operations.

On Wednesday Doctor Yamata could be scheduled to

Doctor Yamata works only on Mondays, Tuesdays, Wednesdays, Fridays, and Saturdays. She performs four different activities—lecturing, operating, treating patients, and conducting research. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. During each week her work schedule must satisfy the following restrictions: She performs operations on exactly three mornings. If she operates on Monday, she does not operate on Tuesday. She lectures in the afternoon on exactly two consecutive calendar days. She treats patients on exactly one morning and exactly three afternoons. She conducts research on exactly one morning. On Saturday she neither lectures nor performs operations.

Which one of the following statements must be true?

Doctor Yamata works only on Mondays, Tuesdays, Wednesdays, Fridays, and Saturdays. She performs four different activities—lecturing, operating, treating patients, and conducting research. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. During each week her work schedule must satisfy the following restrictions: She performs operations on exactly three mornings. If she operates on Monday, she does not operate on Tuesday. She lectures in the afternoon on exactly two consecutive calendar days. She treats patients on exactly one morning and exactly three afternoons. She conducts research on exactly one morning. On Saturday she neither lectures nor performs operations.

If Doctor Yamata operates on Tuesday, then her schedule for treating patients could be

Doctor Yamata works only on Mondays, Tuesdays, Wednesdays, Fridays, and Saturdays. She performs four different activities—lecturing, operating, treating patients, and conducting research. Each working day she performs exactly one activity in the morning and exactly one activity in the afternoon. During each week her work schedule must satisfy the following restrictions: She performs operations on exactly three mornings. If she operates on Monday, she does not operate on Tuesday. She lectures in the afternoon on exactly two consecutive calendar days. She treats patients on exactly one morning and exactly three afternoons. She conducts research on exactly one morning. On Saturday she neither lectures nor performs operations.

Which one of the following is a pair of days on both of which Doctor Yamata must treat patients?

Each of seven judges voted for or else against granting Datalog Corporation's petition. Each judge is categorized as conservative, moderate, or liberal, and no judge is assigned more than one of those labels. Two judges are conservatives, two are moderates, and three are liberals. The following is known about how the judges voted: If the two conservatives and at least one liberal voted the same way as each other, then both moderates voted that way. If the three liberals voted the same way as each other, then no conservative voted that way. At least two of the judges voted for Datalog, and at least two voted against Datalog. At least one conservative voted against Datalog.

If the two moderates did not vote the same way as each other, then which one of the following could be true?

Each of seven judges voted for or else against granting Datalog Corporation's petition. Each judge is categorized as conservative, moderate, or liberal, and no judge is assigned more than one of those labels. Two judges are conservatives, two are moderates, and three are liberals. The following is known about how the judges voted: If the two conservatives and at least one liberal voted the same way as each other, then both moderates voted that way. If the three liberals voted the same way as each other, then no conservative voted that way. At least two of the judges voted for Datalog, and at least two voted against Datalog. At least one conservative voted against Datalog.

Which one of the following must be true?

Each of seven judges voted for or else against granting Datalog Corporation's petition. Each judge is categorized as conservative, moderate, or liberal, and no judge is assigned more than one of those labels. Two judges are conservatives, two are moderates, and three are liberals. The following is known about how the judges voted: If the two conservatives and at least one liberal voted the same way as each other, then both moderates voted that way. If the three liberals voted the same way as each other, then no conservative voted that way. At least two of the judges voted for Datalog, and at least two voted against Datalog. At least one conservative voted against Datalog.

If the three liberals all voted the same way as each other, which one of the following must be true?

Each of seven judges voted for or else against granting Datalog Corporation's petition. Each judge is categorized as conservative, moderate, or liberal, and no judge is assigned more than one of those labels. Two judges are conservatives, two are moderates, and three are liberals. The following is known about how the judges voted: If the two conservatives and at least one liberal voted the same way as each other, then both moderates voted that way. If the three liberals voted the same way as each other, then no conservative voted that way. At least two of the judges voted for Datalog, and at least two voted against Datalog. At least one conservative voted against Datalog.

If exactly two judges voted against Datalog, then which one of the following must be true?

Each of seven judges voted for or else against granting Datalog Corporation's petition. Each judge is categorized as conservative, moderate, or liberal, and no judge is assigned more than one of those labels. Two judges are conservatives, two are moderates, and three are liberals. The following is known about how the judges voted: If the two conservatives and at least one liberal voted the same way as each other, then both moderates voted that way. If the three liberals voted the same way as each other, then no conservative voted that way. At least two of the judges voted for Datalog, and at least two voted against Datalog. At least one conservative voted against Datalog.

Each of the following could be a complete and accurate list of those judges who voted for Datalog EXCEPT

Each of seven judges voted for or else against granting Datalog Corporation's petition. Each judge is categorized as conservative, moderate, or liberal, and no judge is assigned more than one of those labels. Two judges are conservatives, two are moderates, and three are liberals. The following is known about how the judges voted: If the two conservatives and at least one liberal voted the same way as each other, then both moderates voted that way. If the three liberals voted the same way as each other, then no conservative voted that way. At least two of the judges voted for Datalog, and at least two voted against Datalog. At least one conservative voted against Datalog.

If the two conservatives voted the same way as each other, but the liberals did not all vote the same way as each other, then each of the following must be true EXCEPT:

An official is assigning five runners—Larry, Ned, Olivia, Patricia, and Sonja—to parallel lanes numbered consecutively 1 through 5. The official will also assign each runner to represent a different charity—F, G, H, J, and K—not necessarily in order of the runner's names as given. The following ordering restrictions apply: The runner representing K is assigned to lane 4. Patricia is assigned to the only lane between the lanes of the runners representing F and G. There are exactly two lanes between Olivia's lane and the lane of the runner representing G. Sonja is assigned to a higher-numbered lane than the lane to which Ned is assigned.

Which one of the following is a possible assignment of runners to lanes by the charity they represent?

An official is assigning five runners—Larry, Ned, Olivia, Patricia, and Sonja—to parallel lanes numbered consecutively 1 through 5. The official will also assign each runner to represent a different charity—F, G, H, J, and K—not necessarily in order of the runner's names as given. The following ordering restrictions apply: The runner representing K is assigned to lane 4. Patricia is assigned to the only lane between the lanes of the runners representing F and G. There are exactly two lanes between Olivia's lane and the lane of the runner representing G. Sonja is assigned to a higher-numbered lane than the lane to which Ned is assigned.

The lane to which Patricia is assigned must be a lane that is

An official is assigning five runners—Larry, Ned, Olivia, Patricia, and Sonja—to parallel lanes numbered consecutively 1 through 5. The official will also assign each runner to represent a different charity—F, G, H, J, and K—not necessarily in order of the runner's names as given. The following ordering restrictions apply: The runner representing K is assigned to lane 4. Patricia is assigned to the only lane between the lanes of the runners representing F and G. There are exactly two lanes between Olivia's lane and the lane of the runner representing G. Sonja is assigned to a higher-numbered lane than the lane to which Ned is assigned.

If Olivia is assigned to lane 2, which one of the following assignments must be made? Charity Lane

An official is assigning five runners—Larry, Ned, Olivia, Patricia, and Sonja—to parallel lanes numbered consecutively 1 through 5. The official will also assign each runner to represent a different charity—F, G, H, J, and K—not necessarily in order of the runner's names as given. The following ordering restrictions apply: The runner representing K is assigned to lane 4. Patricia is assigned to the only lane between the lanes of the runners representing F and G. There are exactly two lanes between Olivia's lane and the lane of the runner representing G. Sonja is assigned to a higher-numbered lane than the lane to which Ned is assigned.

Which one of the following, is a complete and accurate list of runners each of whom could be the runner representing F?

An official is assigning five runners—Larry, Ned, Olivia, Patricia, and Sonja—to parallel lanes numbered consecutively 1 through 5. The official will also assign each runner to represent a different charity—F, G, H, J, and K—not necessarily in order of the runner's names as given. The following ordering restrictions apply: The runner representing K is assigned to lane 4. Patricia is assigned to the only lane between the lanes of the runners representing F and G. There are exactly two lanes between Olivia's lane and the lane of the runner representing G. Sonja is assigned to a higher-numbered lane than the lane to which Ned is assigned.

If Ned is the runner representing J, then it must be true that

An official is assigning five runners—Larry, Ned, Olivia, Patricia, and Sonja—to parallel lanes numbered consecutively 1 through 5. The official will also assign each runner to represent a different charity—F, G, H, J, and K—not necessarily in order of the runner's names as given. The following ordering restrictions apply: The runner representing K is assigned to lane 4. Patricia is assigned to the only lane between the lanes of the runners representing F and G. There are exactly two lanes between Olivia's lane and the lane of the runner representing G. Sonja is assigned to a higher-numbered lane than the lane to which Ned is assigned.

If Larry represents J, which one of the following could be the assignment of runners to lanes?

A gymnastics instructor is planning a weekly schedule, Monday through Friday, of individual coaching sessions for each of six students—H, I, K, O, U, and Z. The instructor will coach exactly one student each day, except for one day when the instructor will coach two students in separate but consecutive sessions. The following restrictions apply: H's session must take place at some time before Z's session. I's session is on Thursday. K's session is always scheduled for the day immediately before or the day immediately after the day for which O's session is scheduled. Neither Monday nor Wednesday can be a day for which two students are scheduled.

Which one of the following is a pair of students whose sessions can both be scheduled for Tuesday, not necessarily in the order given?

A gymnastics instructor is planning a weekly schedule, Monday through Friday, of individual coaching sessions for each of six students—H, I, K, O, U, and Z. The instructor will coach exactly one student each day, except for one day when the instructor will coach two students in separate but consecutive sessions. The following restrictions apply: H's session must take place at some time before Z's session. I's session is on Thursday. K's session is always scheduled for the day immediately before or the day immediately after the day for which O's session is scheduled. Neither Monday nor Wednesday can be a day for which two students are scheduled.

If K's session is scheduled for Tuesday, then which one of the following is the earliest day for which Z's session can be scheduled?

A gymnastics instructor is planning a weekly schedule, Monday through Friday, of individual coaching sessions for each of six students—H, I, K, O, U, and Z. The instructor will coach exactly one student each day, except for one day when the instructor will coach two students in separate but consecutive sessions. The following restrictions apply: H's session must take place at some time before Z's session. I's session is on Thursday. K's session is always scheduled for the day immediately before or the day immediately after the day for which O's session is scheduled. Neither Monday nor Wednesday can be a day for which two students are scheduled.

Which one of the following must be true?

A gymnastics instructor is planning a weekly schedule, Monday through Friday, of individual coaching sessions for each of six students—H, I, K, O, U, and Z. The instructor will coach exactly one student each day, except for one day when the instructor will coach two students in separate but consecutive sessions. The following restrictions apply: H's session must take place at some time before Z's session. I's session is on Thursday. K's session is always scheduled for the day immediately before or the day immediately after the day for which O's session is scheduled. Neither Monday nor Wednesday can be a day for which two students are scheduled.

Scheduling Z's session for which one of the following days determines the day for which U's session must be scheduled?

A gymnastics instructor is planning a weekly schedule, Monday through Friday, of individual coaching sessions for each of six students—H, I, K, O, U, and Z. The instructor will coach exactly one student each day, except for one day when the instructor will coach two students in separate but consecutive sessions. The following restrictions apply: H's session must take place at some time before Z's session. I's session is on Thursday. K's session is always scheduled for the day immediately before or the day immediately after the day for which O's session is scheduled. Neither Monday nor Wednesday can be a day for which two students are scheduled.

If H's session is scheduled as the next session after U's session, which one of the following could be true about H's session and U's session?

A square parking lot has exactly eight lights—numbered 1 through 8—situated along its perimeter as diagramed below. The lot must always be illuminated in such a way that the following specifications are met: At least one of any three consecutively numbered lights is off. Light 8 is on. Neither light 2 nor light 7 is on when light 1 is on. At least one of the three lights on each side is on. If any side has exactly one of its three lights on, then that light is its center light. Two of the lights on the north side are on.

Which one of the following could be a complete and accurate list of lights that are on together?

A square parking lot has exactly eight lights—numbered 1 through 8—situated along its perimeter as diagramed below. The lot must always be illuminated in such a way that the following specifications are met: At least one of any three consecutively numbered lights is off. Light 8 is on. Neither light 2 nor light 7 is on when light 1 is on. At least one of the three lights on each side is on. If any side has exactly one of its three lights on, then that light is its center light. Two of the lights on the north side are on.

Which one of the following lights must be on?

A square parking lot has exactly eight lights—numbered 1 through 8—situated along its perimeter as diagramed below. The lot must always be illuminated in such a way that the following specifications are met: At least one of any three consecutively numbered lights is off. Light 8 is on. Neither light 2 nor light 7 is on when light 1 is on. At least one of the three lights on each side is on. If any side has exactly one of its three lights on, then that light is its center light. Two of the lights on the north side are on.

If light 1 is off, which one of the following is a light that must also be off?