# LSAT Logic Games : Two-Variable

## Example Questions

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### Example Question #1 : Solving Two Variable Logic Games

A photographer is hanging six portraits on the wall in a straight line. The portraits are of six family members: Lily, Mildred, Nancy, Owen, Peter and Quentin. The order in which the portraits are hung must conform to the following restrictions:

Mildred's portrait must be either first or last

There must be exactly one portrait between Nancy and Quentin

Nancy's portrait must come after Lily's but before Quentin's

Which of the following is an acceptable order in which the portraits could be hung?

Mildred, Nancy, Lily, Quentin, Owen, Peter

Lily, Quentin, Peter, Nancy, Owen, Mildred

Lily, Nancy, Quentin, Peter, Owen, Mildred

Mildred, Lily, Owen, Nancy, Peter, Quentin

Lily, Mildred, Nancy, Owen, Quentin, Peter

Mildred, Lily, Owen, Nancy, Peter, Quentin

Explanation:

Any option without Mildred being first or last is immediately taken out. Any option without one space between Nancy and Quentin is taken out, and any option that has Quentin appearing before Nancy is taken out. Any option that doesn't have Lily appearing before Nancy is taken out, leaving only the correct answer as an option.

### Example Question #2 : Two Variable

A photographer is hanging six portraits on the wall in a straight line. The portraits are of six family members: Lily, Mildred, Nancy, Owen, Peter and Quentin. The order in which the portraits are hung must conform to the following restrictions:

Mildred's portrait must be either first or last

There must be exactly one portrait between Nancy and Quentin

Nancy's portrait must come after Lily's but before Quentin's

If Nancy's portrait is second, each of the following must be true EXCEPT:

Peter is third

Mildred is last

Lily is first

Owen is somewhere after Nancy

Quentin is fourth

Peter is third

Explanation:

If Nancy is second, we can immediately put Quentin in the fourth spot. Since Lily must come before Nancy, her portrait must be first. With Lily occupying the first spot, Mildred then must be last. We are left with Owen and Peter to fill out the thrid and fifth spots, in either order. Therefore, every answer given must be true EXCEPT that Peter is third - he could be, but he doesn't have to be.

### Example Question #3 : Two Variable

A photographer is hanging six portraits on the wall in a straight line. The portraits are of six family members: Lily, Mildred, Nancy, Owen, Peter and Quentin. The order in which the portraits are hung must conform to the following restrictions:

Mildred's portrait must be either first or last

There must be exactly one portrait between Nancy and Quentin

Nancy's portrait must come after Lily's but before Quentin's

If Quentin must come before Peter, how many possible orders are there in which the portraits can be hung?

Three

None

Four

Two

One

Two

Explanation:

This gives us a new rule, namely that Quentin must come before Peter. We already know that Nancy must appear exactly two spots before Quentin. We also know that Lily has to come somewhere before Nancy. Our complete new rule states that Lily must come somewhere before Nancy, who must appear exactly two spots before Quentin, who must appear somewhere before Peter. When we add all of these up (including the blank spot between Nancy and Quentin) we have already filled out five of the six possible spots. When we take into account that Mildred must be in either the first or last spot, we are left with two options for this super block. Lily, starting off the block, can either go in the first spot and Mildred in the last, or in the second spot with Mildred in the first. In both scenarios Owen fills in the missing spot between Nancy and Quentin. Therefore, there are only two possible solutions that adhere to all of the given rules and the new added rule.

### Example Question #1 : Solving Two Variable Logic Games

A gym teacher wants his 6 students to line up in height order, from shortest to tallest. Corrin and Theresa are girls. Ben, Jonathan, Will, and Dan are the boys.

- Will is not the tallest or the shortest.

- No girl is taller than Jonathan

- Dan is shorter than Corrin, but taller than Theresa.

- Ben is the tallest

Which of the following must be true?

If Jonathan is 3rd tallest, Will is 4th tallest.

If Will is 5th in line, Dan is 2nd in line.

If Dan is 2nd in line, Corrin must be 3rd.

If Theresa is shortest, Corrin is 2 spots behind her.

If Ben is tallest, Corrin is the 3rd tallest.

If Will is 5th in line, Dan is 2nd in line.

Explanation:

This can be answered by process of elimination. Ben being in 6th doesn't effect anyone, as he is always there. If Jonathan is 3rd tallest, the only one who can be the 5th spot is Will - thus, that answer is wrong. Theresa must be the first person in line, so she doesn't affect any positioning. Thus, Corrin could be 3rd, or 4th in line, despite Theresa's position. Finally, if Dan is 2nd in line, Corrin is not guaranteed to be at the 3rd position. Will can be at the third position, pushing Corrin to the 4th.

### Example Question #1 : Solving Two Variable Logic Games

A librarian is organizing seven categories of books on seven shelves, numbered one to seven from top to bottom. The categories are art, botany, calculus, food, sports, theology, and zoology. The librarian places one category of books on each shelf, and does so according to the following rules:

Food must always be on either the top or bottom shelves

Art must be directly above or below calculus

Theology is never on the top or the bottom shelves

Calculus and botany must be separated by exactly one category

Which of the following is a possible order of the book categories on the shelves, from top to bottom?

S, C, A, B, T, Z, F

F, C, A, B, S, Z, T

F, A, C, B, T, S, Z

A, C, S, Z, B, T, F

S, B, T, C, A, F, Z

S, C, A, B, T, Z, F

Explanation:

Answers can be quickly eliminated when F is not the first or last letter or when T is the first or last letter.  Other answers can be eliminated when A is not directly next to C.  Finally, there must be precisely one letter between B and C.  Remember, A can be the letter that separates B and C.

### Example Question #1 : Two Variable

A librarian is organizing seven categories of books on seven shelves, numbered one to seven from top to bottom. The categories are art, botany, calculus, food, sports, theology, and zoology. The librarian places one category of books on each shelf, and does so according to the following rules:

Food must always be on either the top or bottom shelves

Art must be directly above or below calculus

Theology is never on the top or the bottom shelves

Calculus and botany must be separated by exactly one category

If calculus is assigned the spot directly below the top shelf, then each of the following could be true EXCEPT

Theology is one shelf above botany

Sports is directly below food

Food is on the top shelf

Zoology is on a shelf directly above or directly below sports

Art is one shelf above botany

Sports is directly below food

Explanation:

Sports could be directly above food if food is on the bottom shelf, but it could never be below food because food must either be on the top shelf or the bottom shelf.  No category can be directly below food when it is on the bottom shelf and we know that calculus is directly below the top shelf.

### Example Question #7 : Solving Two Variable Logic Games

A librarian is organizing seven categories of books on seven shelves, numbered one to seven from top to bottom. The categories are art, botany, calculus, food, sports, theology, and zoology. The librarian places one category of books on each shelf, and does so according to the following rules:

Food must always be on either the top or bottom shelves

Art must be directly above or below calculus

Theology is never on the top or the bottom shelves

Calculus and botany must be separated by exactly one category

When calculus is on the bottom shelf, which of the following could be true, but is not required to be true?

Botany is on a shelf above art

Zoology is on the fifth shelf

Food is on the top shelf

Zoology is on a shelf directly above theology

Art is on the shelf directly below sports

Zoology is on a shelf directly above theology

Explanation:

When calculus is on the bottom shelf, food must be on the top shelf and botany and art must occupy the fifth and sixth shelves.  The remaining categories, however, can be in any order on shelves two, three, and four.  Zoology could be on the shelf directly above theology, but it does not have to be.

### Example Question #8 : Solving Two Variable Logic Games

A librarian is organizing seven categories of books on seven shelves, numbered one to seven from top to bottom. The categories are art, botany, calculus, food, sports, theology, and zoology. The librarian places one category of books on each shelf, and does so according to the following rules:

Food must always be on either the top or bottom shelves

Art must be directly above or below calculus

Theology is never on the top or the bottom shelves

Calculus and botany must be separated by exactly one category

If a condition is added that sports must always be on a shelf between botany and calculus, and if all other conditions remain the same, which of the following cannot be true?

Botany is on the sixth shelf and art is on the third

Zoology is on the second shelf and calculus is on the fourth shelf

Theology is on the second shelf

Calculus is on the fourth shelf

Zoology is on the last shelf and sports are on the fifth

Zoology is on the second shelf and calculus is on the fourth shelf

Explanation:

When calculus is on the fourth shelf, botany must be on the second or sixth.  Since zoology is on the second shelf, botany must be on the sixth.  Sports must be on the fifth shelf to be between botany and calculus, and art must be on the second to be adjacent to calculus, which leaves only the top or bottom for theology.

### Example Question #9 : Solving Two Variable Logic Games

A media company is determining the lineup for its programming tonight. There are five hour long shows – P, Q, R, S, T – that must be aired one after another from 6:00 to 11:00. Each show must be paired with one of three newscasters – Adrian, Brett, Calvin – subject to the following conditions:

Each newscaster must host at least one show.
Adrian cannot host a show after 9:00.
There must be exactly two shows in between Adrian’s first show and Calvin’s first show.
Q is aired before R.
R is aired before both S and T.

Which one of the following could be an accurate and complete list of the order the shows air along with the newscasters assigned to host them?

Explanation:

The incorrect answers all violate one of the stated conditions:

(P: Adrian, Q: Brett, R: Adrian, S: Brett, T: Calvin) - Adrian's first show must have exactly two shows in between it and Calvin's first show. This has three.

(Q: Adrian, R: Brett, T, Brett, P: Calvin, S: Adrian) - Adrian cannot have any shows after 9:00; the last slot would be 10:00-11:00, a violation of the rule.

(P: Brett, Q: Adrian, S: Adrian, R: Brett, T: Calvin) - R is aired before both S and T; it is aired after S in this case.

(P: Adrian, Q: Adrian, R: Adrian, S: Calvin, T: Calvin) - All of the newcasters must host at least one show, so Brett's absence here is a violation.

The correct answer does not violate any of the stated conditions.

### Example Question #1 : Two Variable

A media company is determining the lineup for its programming tonight. There are five hour long shows – P, Q, R, S, T – that must be aired one after another from 6:00 to 11:00. Each show must be paired with one of three newscasters – Adrian, Brett, Calvin – subject to the following conditions:

Each newscaster must host at least one show.
Adrian cannot host a show after 9:00.
There must be exactly two shows in between Adrian’s first show and Calvin’s first show.
Q is aired before R.
R is aired before both S and T.

If P is aired from 6:00-7:00 with Brett as the host, which of the following must be true?

S is aired from 10:00-11:00

Calvin hosts the 10:00-11:00 show.

Brett hosts the 8:00-9:00 show.

S is aired from 9:00-10:00.

Calvin hosts the 10:00-11:00 show.

Explanation:

Since we are given Brett as the host of the first show, this means that Adrian must host the 7:00-8:00 show and Calvin must host the 10:00-11:00 show in order to put exactly two shows in between their initial shows. Moreover, the shows in between must be hosted by Adrian or Brett because Calvin must have his first performance in the 10:00-11:00 slot. This gives us the following order:

6:00-7:00: P: Brett
10:00-11:00: Calvin

If one combines the last two rules, the following order in which the shows air is created:

Q - R - (S/T)

Applying this to the above model, we get:

6:00-7:00: P: Brett