# LSAT Logic Games : Solving three-variable logic games

## Example Questions

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### Example Question #1 : Three Variable

A baker is making three pizzas, one at a time, each with two toppings. The baker has six available toppings--anchovies, bacon, mushrooms, peppers, sausage, tomatoes. No topping can be put on more than one pizza. The pairings of toppings must conform to the following rules:

Anchovies cannot be paired with peppers

Mushrooms and tomatoes must be on the same pizza

Sausage must be on the second pizza if mushrooms are on the first

Peppers must be on a pizza made after the pizza with sausage

Which of the following is a possible ordering of the pizzas and toppings?

Mushrooms and tomatoes; sausage and peppers; bacon and anchovies

Sausage and bacon; mushrooms and tomatoes; anchovies and peppers

Sausage and anchovies; mushrooms and tomatoes; peppers and bacon

Sausage and tomatoes; peppers and anchovies; mushrooms and bacon

Peppers and bacon; mushrooms and tomatoes; anchovies and sausage

Sausage and anchovies; mushrooms and tomatoes; peppers and bacon

Explanation:

If peppers must be on a pizza made after the sausage pizza, then peppers can never be on the first pizza or the same pizza as sausage.  We can also easily eliminate any option in which tomatoes and mushrooms are not paired.  Remember, sausage is only required to be on the second pizza if mushrooms are on the first.

### Example Question #2 : Three Variable

A baker is making three pizzas, one at a time, each with two toppings. The baker has six available toppings--anchovies, bacon, mushrooms, peppers, sausage, tomatoes. No topping can be put on more than one pizza. The pairings of toppings must conform to the following rules:

Anchovies cannot be paired with peppers

Mushrooms and tomatoes must be on the same pizza

Sausage must be on the second pizza if mushrooms are on the first

Peppers must be on a pizza made after the pizza with sausage

If sausage is on the second pizza, which of the following is a complete list of toppings that must be on the third pizza?

Bacon and peppers

Bacon

Mushrooms and tomatoes

Peppers

Peppers and tomatoes

Bacon and peppers

Explanation:

Since peppers must be on a pizza made after the pizza with sausage, peppers must be on the third pizza.  Now, since the second and third pizzas already have one topping each, and since mushrooms and tomatoes must be on the same pizza, they must be on the first pizza.  As a result, anchovies must be on the second pizza because they cannot be on the same pizza as peppers.  The only remaining spot for bacon is on the third pizza.  Therefore, both bacon and peppers must be on the third pizza.

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

A baker is making three pizzas, one at a time, each with two toppings. The baker has six available toppings--anchovies, bacon, mushrooms, peppers, sausage, tomatoes. No topping can be put on more than one pizza. The pairings of toppings must conform to the following rules:

Anchovies cannot be paired with peppers

Mushrooms and tomatoes must be on the same pizza

Sausage must be on the second pizza if mushrooms are on the first

Peppers must be on a pizza made after the pizza with sausage

Instead of three pizzas, the baker makes four.  Anchovies and bacon are each used on two separate pizzas.  All other conditions are the same.  If anchovies are on the second and fourth pizzas, each of the following could be true EXCEPT

Mushrooms and tomatoes are on the first pizza

Peppers are on the third pizza

Bacon and sausage are on different pizzas

Anchovies and bacon are on the same pizza

Anchovies and sausage are on different pizzas

Anchovies and sausage are on different pizzas

Explanation:

In this scenario, we actually know the toppings on all four pizzas.

Since anchovies are on the second and fourth pizzas, we know that peppers cannot be on the second and fourth pizzas based on our first condition.  Also, peppers have to come after sausage, based on the fourth condition, which means they cannot be first.  Thus, peppers MUST be on the third pizza.

The second condition states that mushrooms and tomatoes MUST be on the same pizza.  At this point, since anchovies are on two pizzas, and peppers on another, the only possible pizza for both mushrooms and tomatoes is the first pizza.  The first pizza is now fully topped.

The third condition states that sausage must be on the second pizza if mushrooms are on the first pizza.  Therefore, sausage must be the second topping on the second pizza.  The second pizza is fully topped.

There are only two spots available, so bacon becomes the second topping on both the third and the fourth pizzas.  So our order must be:

MT, AS, PB, AB

### Example Question #2 : Solving Three Variable Logic Games

A college advisor is scheduling students for six time slots, consecutively. The students she must schedule are Larissa, Melinda, Nick, Oscar, Patricia, and Quinn. Two of them are seniors and the rest are juniors. The scheduling must conform to the following conditions:

A senior must be immediately preceeded by a junior

Larissa's meeting must be before Melinda's meeting

Nick's meeting must be before Quinn's meeting

Patricia is a junior, and she must be schedued either first or last

The third meeting scheduled is with a junior

Oscar cannot be scheduled first unless Nick is scheduled third

Which of the following must be true?

The fifth student is a junior

The second student is a senior

The sixth student is a senior

The first student is a junior

The fourth student is a senior

The first student is a junior

Explanation:

This is an inference we could have made just from the initial set up of this game. Since seniors must always be preceeded by juniors, the first spot must always be a junior. All of the other scenarios are possible and most have been seen in other set ups for other questions.

### Example Question #3 : Solving Three Variable Logic Games

A creative writing professor is creating a set list for a poetry reading. She is choosing five poems from those written by eight students - Alan, Belle, Charlie, Dorian, Ernest, Xue, Yardley, and Zack. The poem's chosen and the order in which they are presented must conform to the following restrictions:

If Alan is chosen, Belle is also chosen

If Charlie is chosen, Dorian is not chosen

Ernest is chosen if and only if Xue is chosen

If Belle and Yardley are both chosen, Belle must read before Yardley

If Zack is chosen he must read first

If Charlie and Alan are both chosen, Charlie must read before Alan

If Zack is reading first and Dorian is reading last, which of the following could be a list of the students reading second, third and fourth, respectively?

Ernest, Xue, Belle

Yardley, Belle, Alan

Belle, Xue, Yardley

Xue, Ernest, Alan

Ernest, Charlie, Xue

Ernest, Xue, Belle

Explanation:

If Dorian is reading we know Charlie cannot read, so any answer that includes him can be eliminated. Any answer that includes either Ernest or Xue without the other can also be eliminated. Any answer that includes Alan and not Belle is also eliminated. Any answer that includes Belle and Yardley and does not have Belle reading first is also eliminated, leaving only the correct answer.

### Example Question #4 : Solving Three Variable Logic Games

A college advisor is scheduling students for six time slots, consecutively. The students she must schedule are Larissa, Melinda, Nick, Oscar, Patricia, and Quinn. Two of them are seniors and the rest are juniors. The scheduling must conform to the following conditions:

A senior must be immediately preceeded by a junior

Larissa's meeting must be before Melinda's meeting

Nick's meeting must be before Quinn's meeting

Patricia is a junior, and she must be schedued either first or last

The third meeting scheduled is with a junior

Oscar cannot be scheduled first unless Nick is scheduled third

If Oscar is scheduled for the first meeting, all of the following could be true EXCEPT?

Melinda is fourth and is a senior

Patricia is sixth and is a junior

Larissa is second and is a junior

Melinda is fifth and is a senior

Quinn is fourth and is a junior

Larissa is second and is a junior

Explanation:

This question gives us specific new information, so we can go ahead and diagram all possibilities to see which of the answers could be true, and which one cannot. If Oscar is first, we immediately put Nick third because of the conditional. Since Oscar is in the first spot Patricia must be sixth. We know now that Patricia and Nick are juniors, due to the rules. We also know that Oscar is a junior - since  every senior must be preceeded by a junior, the first spot cannot be a senior. Since Nick and Larissa have to come before Quinn and Melinda respectively, We have to put Larissa in the second spot. Melinda and Quinn can rotate between the fourth and fifth spots. We still need to assign the two seniors and one more junior. In order to abide by the rules, there are only two ways this can pan out. The first is : Junior, Senior, Junior, Senior, Junior, Junior. The second is: Junior, Senior, Junior, Junior, Senior, Junior. Therefore the only spots that are undetermined as far as whether they are a junior or a senior are the fourth and fifth spots. Therefore, the only possibility here that could never work is that Larissa is in the second spot and is a junior, since we know that spot must be a senior.

### Example Question #7 : Three Variable

A creative writing professor is creating a set list for a poetry reading. She must chose five poems from those written by eight students - Alan, Belle, Charlie, Dorian, Ernest, Xue, Yardley, and Zack. The poems chosen and the order in which they are presented must conform to the following restrictions:

If Alan is chosen, Belle is also chosen

If Charlie is chosen, Dorian is not chosen

Ernest is chosen if and only if Xue is chosen

If Belle and Yardley are both chosen, Belle must read before Yardley

If Zack is chosen he must read first

If Charlie and Alan are both chosen, Charlie must read before Alan

Which of the following is a complete and accurate possible set list?

Belle, Zack, Yardley, Ernest, Xue

Zack, Belle, Charlie, Yardley, Alan

Charlie, Xue, Belle, Yardley, Alan

Charlie, Alan, Ernest, Xue, Yardley

Alan, Yardley, Charlie, Belle, Dorian

Zack, Belle, Charlie, Yardley, Alan

Explanation:

This question can be answered by eliminating incorrect answers based on rule violations. Any answer in which Zack appears anywhere but first is elminated. Any answer that includes Alan without Belle is eliminated. Any answer that features Yardley performing before Belle is eliminated. Any answer that includes Xue without Ernest (or vice versa) is eliminated, leaving only the correct answer.

### Example Question #5 : Solving Three Variable Logic Games

A creative writing professor is creating a set list for a poetry reading. She is choosing five poems from those written by eight students - Alan, Belle, Charlie, Dorian, Ernest, Xue, Yardley, and Zack. The poem's chosen and the order in which they are presented must conform to the following restrictions:

If Alan is chosen, Belle is also chosen

If Charlie is chosen, Dorian is not chosen

Ernest is chosen if and only if Xue is chosen

If Belle and Yardley are both chosen, Belle must read before Yardley

If Zack is chosen he must read first

If Charlie and Alan are both chosen, Charlie must read before Alan

If Dorian and Xue are NOT chosen, each of the following must be true EXCEPT:

Explanation:

If Xue is not chosen, Ernest also must not be chosen. This means our group consists of Zack, Charlie, Alan, Belle and Yardley. If Zack is in a group, he must be first. If Charlie and Alan are both chosen, Charlie must come before Alan. In this case the first available spot is second, so Alan cannot go second. Similarly, because of the rule about Belle and Yardley, Yardley cannot go second either. Since Yardley has to come after Belle, Belle cannot go last. The only possibility within these answers is Alan reading third. The order in this case would be: Zack, Charlie, Alan, Belle, Yardley.

### Example Question #9 : Three Variable

A college advisor is scheduling students for six time slots, consecutively. The students she must schedule are Larissa, Melinda, Nick, Oscar, Patricia, and Quinn. Two of them are seniors and the rest are juniors. The scheduling must conform to the following conditions:

A senior must be immediately preceeded by a junior

Larissa's meeting must be before Melinda's meeting

Nick's meeting must be before Quinn's meeting

Patricia is a junior, and she must be schedued either first or last

The third meeting scheduled is with a junior

Oscar cannot be scheduled first unless Nick is scheduled third

Which of the following is a complete and accurate possible schedule for the advisor's meetings?

Patricia, Larissa, Melinda, Quinn, Oscar, Nick

Patricia, Nick, Oscar, Quinn, Melinda, Larissa

Oscar, Larissa, Nick, Quinn, Melinda, Patricia

Oscar, Quinn, Nick, Larissa, Patricia, Melinda

Oscar, Nick, Larissa, Quinn, Melinda, Patricia

Oscar, Larissa, Nick, Quinn, Melinda, Patricia

Explanation:

This is a typical "grab a rule" type question; we can eliminate each wrong answer choice by going through each of the rules. Any answer in which Patricia is not first or last can be eliminated. Then any answer that does not feature Larissa before Melinda and Nick before Quinn can be eliminated. Finally an answer that breaks the conditional and has Oscar in the first slot without Nick in the third slot is eliminated, leaving only the correct answer.

### Example Question #10 : Three Variable

A college advisor is scheduling students for six time slots, consecutively. The students she must schedule are Larissa, Melinda, Nick, Oscar, Patricia, and Quinn. Two of them are seniors and the rest are juniors. The scheduling must conform to the following conditions:

A senior must be immediately preceeded by a junior

Larissa's meeting must be before Melinda's meeting

Nick's meeting must be before Quinn's meeting

Patricia is a junior, and she must be schedued either first or last

The third meeting scheduled is with a junior

Oscar cannot be scheduled first unless Nick is scheduled third

If Larissa and Nick are juniors scheduled third and fourth respectively, which of the following could be true?

Quinn is fifth and is a senior

Melinda is second and is a senior

Oscar is second and is a junior

Patricia is last and is a junior

Oscar is last and is a junior