ANALYTICAL SECTION:
LOGIC GAMES

This section is broken into 7 parts

I. INTRODUCTION
II. STRUCTURE OF A LOGIC GAME
III. GENERAL STRATEGY FOR WORKING LOGIC GAMES
IV. GAME TYPE 1: ORDERING GAMES

V. GAME TYPE 2: CHARACTERISTIC GAMES
VI. GAME TYPE 3: GROUPING GAMES
VII. GAME TYPE 4: NETWORKING GAMES
VIII. GAME TYPE 5: SPATIAL
IX. GAME TYPE 6: MAPS

I. INTRODUCTION

The Analytical Section may at first seem like the most difficult of the three parts of the LSAT , if only because the material in this section is usually the least familiar to students. Unlike the math and verbal sections, most people preparing for the LSAT have never before been tested for their ability to solve the types of problems found in this section. Further, most of the skills and strategies you will need to score well on the analytical section are probably not anything that you would have learned in your college courses. That being said, the problems in the analytical portion of the exam are probably the easiest on which to improve once you learn some basic strategies.

The previous chapter detailed tactics for answering one type of question found in the analytical section, the critical reasoning questions which require analysis of an argument. This chapter will address the other type of question found in this section, the Logic Games. In the sixty-minute analytical section, there are usually 3-6 Logic Games, each game consisting of anywhere from three to eight questions. Thus, each game of multiple questions counts a lot towards your score on this section.

The first portion of this chapter will present an overview of the structure of Logic Games, and teach you how to approach game problems by critically reading the information in the problem and concisely and accurately summarizing the rules according to formal logic. Learning how to read precisely, and training yourself to know what inferences you can make from the information given without making invalid assumptions, are instrumental skills needed to do well on these problems. The next portion of the chapter will describe the four main types of games, and detail how to systematically solve these problems through the effective use of diagrams. Through learning the skills of critical reading, precise rule summary, and accurate and informative diagramming, you will be able to solve any of the Logic Games you encounter on the LSAT .

II. STRUCTURE OF A LOGIC GAME

Logic games are comprised of three parts: the initial scenario or set-up of the game, the list of conditions or rules that apply to the game, and then the set of questions.

The scenario describes the basic situation or task that needs to be solved. For example, the task might be to figure out the order of people standing in a line for a concert, or to schedule meetings for a group of people with various times of availability, or to divide a large group into smaller groups, such as teams or committees. The scenario will also introduce the subjects, or elements, that you will be asked to relate.

The scenario is immediately followed by the conditions, or rules of the game. These rules describe relationships among the elements introduced in the scenario. For example: Maria will go to the picnic only if her cousin Leo does not attend, or each basketball team's roster of five players must include at least two women players, or plaid pants can be worn on Tuesdays or Fridays, but not both days. There may be anywhere from three to six specific conditions for each game, and each one begins on a separate line of text. The information provided by the conditions may describe a fixed relationship: in a line of seven people, Lucas is the fourth person; or the condition may describe a variable relationship: in designing a vegetable garden, the carrots must be planted within two rows of the peas. A variable condition, by definition, can be satisfied with multiple arrangements.

The scenario and the conditions will remain on the screen for all of the questions pertaining to that game (however, you do see only one of the questions at a time). There are three types of questions in Logic Games. The first type is the most basic, and asks you to reach a conclusion about how the task can be solved based on the information given to you in the original scenario and the conditions. The second type of question provides an additional condition to be considered in figuring out how to solve the task. The third type of question is one that alters or negates either some part of the original scenario or one of the conditions. The critical thing to remember about the latter two types of questions is that the new or altered condition presented in these kinds of questions pertains ONLY to the specific question in which the new information is provided. The information from one question is NOT applicable to subsequent ones; each question must be considered independently. Only the information from the scenario and original conditions can be applied universally to each question within the game (unless specifically modified as in the third type of questions).

III. GENERAL STRATEGY FOR WORKING LOGIC GAMES

The latter portion of this chapter will describe the four most common game types, along with specific strategies and diagramming techniques for solving each type. However, to start out, we will begin by summarizing the basic strategy that you can apply to any logic game, and then we will look at each of these steps in some detail, as we work through a sample game using this basic strategy.

Summary of general strategy:

3. Rewrite each condition on your scrap paper in symbolic form.

4. Check over what you have written to see if there is any additional information about relationships that can be deduced after considering the conditions together.

5. Construct a diagram to incorporate all of the information in a general picture.

6. Read the question; if any new information is provided, summarize that condition as well and add that to the existing diagram.

7. Use the diagram to answer the question.

Now that we have summarized the general strategy, let's look at each of these steps in greater detail.

What is the task that you are being asked to solve? (Later in this chapter, you will also learn to distinguish between different types of game problems in order to apply game-specific strategies.) What kinds of questions do we expect will be asked for this particular game?

You must pay particular attention to what each condition states directly, as well as consider what the condition may imply. It is critical that you pay attention to keywords that describe or limit relationships. Key words may position elements of a puzzle either in a fixed position, such as by stating that the acrobats are the fourth circus act to perform, or in a variable way, such as by stating that Bob will play chess after Julia. Other key words to note are: all, some, none, always, and, or, may, might, never, sometimes, only, preceding, following, immediately, at least, exactly, must be, cannot be, or if…then. Pay close attention to whether the condition is a requirement, such as: A group must contain three members; a prohibition: No group may contain three members; an allowance: A group may contain from three to five members, or a conditional: If any group contains three members, then no other group may contain five members.

These words must be read and considered literally and precisely, without making any invalid assumptions. For example, if we are told that six people are sitting in the front row of a movie theater, and Shondelle is sitting to the right of Linda, we must not assume that Shondelle is sitting next to Linda on her right, only that she is somewhere to the right of Linda, and there might be others seated in between them.

3. Rewrite each condition on your scrap paper in symbolic form.
One of the major difficulties people have with Logic Games is keeping track of so many conditions. This may be an even greater challenge now that the LSAT is a computer-based, rather than a paper-based exam. Because of the time constraints, you are not able to keep rereading through all the conditions. It is critical that you become proficient at learning to rewrite the conditions onto your scrap paper in abbreviated, symbolic form and to do this in a way that preserves the accuracy of the information provided.

There is no single way to go about symbolizing statements, and you should experiment with different techniques as you work through the sample exams. Here are some techniques that we find helpful:

Write down a complete roster of the elements or subjects of the game. Although the scenario will clearly identify the elements of the game (people, events, times, objects, etc.), the individual conditions will not necessarily describe all of the elements. Because of this, it can be easy to lose track of how many elements there are unless you create a roster. A good technique is to use capital letters in place of the names of the elements, each letter chosen being the first letter of the word. (Luckily, on the LSAT the names of the elements almost always start with different letters of the alphabet.) For example, if the game is to plan a daily course schedule of six classes: art, biology, chemistry, economics, history, and sociology, the roster can be represented as

CLASSES: A, B, C, E, H, S

Next, rewrite each condition onto your scrap paper, starting with any information present in the scenario, incorporating symbols to represent relationships. Mathematical symbols can be useful for this, such as <, >, =, etc. If we are told that in a group of six children whose ages range from 5 to 10, Maria and Alex are younger than Irina, we can summarize this as M, A < I. In this case, we are using the less than symbol to represent the relative ages of the children; Maria and Alex are younger than Irina has become M, A < I. It is also useful to note that we could also break this into two separate statements, M < I and A < I. Of course, whenever we are told of a greater than or less than relationship, we automatically know something about the reverse of that relationship. That is, not only do we know that M < I, and A < I, but we also know I > M and I > A. Likewise, we can utilize the < and > symbols if we are sorting the elements of a game according to any another criteria, such as by size, height, or mass. For example, of seven cities, Detroit and Chicago have a greater population than Orlando, becomes D, C > O.

What other kinds of symbols are needed? One important concept often appearing in conditions is a limit that excludes an element from a certain position, such as:

Of five people meeting at a restaurant for lunch, Lucinda cannot be the first person to arrive.

In this case, we want to symbolize that Lucinda is not first. We could write this as L ¹ 1. Another common way to write this is L ^ 1, with the ^ representing "not". Or you could simply write L not 1.

Another important concept that needs to be represented symbolically is a space, or unknown. You might be told that two people are seated between Marco and Elena. This could be represented as M _ _ E, or M sp. sp. E, where sp. represents space. Keep in mind that for an example like this, where we are told how many people are between two other people, this only tells us about the spacing between M and E, and it does not tell us anything about who, Marco or Elena, is on the left or the right (or further in front or back). Therefore, we could also represent this information as E _ _ M.
This is an important concept to keep in mind when interpreting the conditions of the Logic Games. Do not assume that because Marco's name is first that this implies anything about the order. Always interpret the conditions of Logic Games strictly and literally.

Keep in mind that you can also choose to devise your own symbols. Just make sure that whatever symbols you choose are meaningful to you.

Another type of condition that you will need to be able to symbolize for game problems is the if…then condition. If…then conditions provide provisional information about the relationships between the elements of a game.

The following is an example of an if…then condition:

Of ten kids who are auditioning, five kids must be selected to sing the national anthem at the start of the baseball game.

If Luis is selected, then Antonio cannot be selected.

What does this condition mean? Obviously, we know that if Luis is selected to sing, Antonio is not selected.

We can write this as: if L, then not A. (Or: if L, then ^ A.)

Are there other conclusions that we can draw from this information? Yes, we also know that if Antonio is selected to sing, Luis must not have been selected.

We can write this as: if A, then not L. (Or: if A, then ^L.)

Being able to come up with this second conclusion from a logical statement (called the contrapositive) is an extremely useful skill that you should work on when summarizing the conditions for a game problem.

Now, think about what happens if Luis is not selected. Does the condition tell us what happens to Antonio in this case? Will he be selected? We don't know. Likewise, if Antonio is not selected, we do not know whether or not Luis will be selected. The condition's requirement is that Luis and Antonio cannot both be selected to sing the national anthem, though it does not say anything about whether either of them must be selected to sing. You must pay particular attention to what you can infer from an if…then condition, and not make any invalid assumptions. Another factor to keep in mind when faced with if…then conditions is that they may provide additional information in combination with other conditions.

Let's look at a game problem and try to use the above guidelines to symbolize the information:

The owner of a pizza parlor is planning the work schedule for his six employees, Edgar, Reuben, Teresa, Jan, Nina, and Morris.

At least two and not more than four employees must work each shift.
Edgar and Reuben cannot work together.
Morris will only work if Jan is also working.
Teresa and Reuben can work together only if Morris is working also.
If Jan is working, then either Morris or Reuben must be working.

Okay, let's start by identifying what the central task is. In this case, we are being asked to make a work schedule for six employees.

The next step is to make the roster of the elements of the game. In this case, the elements we are asked to arrange are the six employees, so our roster will look something like this:

Employees: E, R, T, J, N, M

Note: It is critical to make the roster rather than just skipping ahead to summarize the conditions. Often, as in this case, one or more of the elements of the game will not be mentioned in the conditions (Nina is not mentioned in the conditions). By creating a roster, you will ensure that you won't lose track of all the elements.

Now we can proceed by summarizing the information provided in the conditions:

The first statement tells us that at least two and not more than four employees must work each day. We can write this in an abbreviated form as:

# of emp. = 2, 3, or 4.

The second statement tells us that Edgar and Reuben cannot work together. We can summarize this as:

If E, then no R; if R, then no E.

The third statement tells us that Morris will only work if Jan is also working. We can summarize this as:

If M, then J.

Also: If ^J, then ^M.

(Can we write this as if J, then M? No. Jan can work without Morris working, but Morris can only work if Jan is working. Knowing the difference between the validity of these two statements is absolutely critical to solving Logic Games.)

The next statement tells us that Teresa and Reuben can work together only if Morris is working also. We can summarize this as:

T and R only if M.

(Do we know if T can work with M if R is not working? Can M work with R if T is not working? We cannot answer these questions from this condition alone. Only that T and R can only work together is M is also working.)

The final statement tells us that if Jan is working, then either Morris or Reuben must be working. We can summarize this condition as:

If J, then M or R.

What other information can we derive from this? Again, you can write the contrapositive of this statement:

If ^M and ^R then ^J.

Let's look at the summary we have created thus far:

Employees: E, R, T, J, N, M

num. of emp. = 2, 3, or 4
If E, then no R, if R, then no E
If M, then J, if ^J, then ^M
T and R only if M
If J, then M or R, if ^M and ^R then ^J

By working at the practice games and learning how to precisely and accurately summarize the information given to you in the conditions, you will be well on your way to solving the Logic Games.

4. Check over what you have written to see if there is any additional information about relationships that can be deduced after considering the conditions together.
Once you have summarized each of the conditions individually, you should consider them together. Look for conditions that share an element. These will be the conditions that may be able to be combined. If we put these conditions together, we can see if any new information is provided. Let's again consider the game of employees at the pizza parlor:

The owner of a pizza parlor is planning the work schedule for the week for his six employees, Edgar, Reuben, Teresa, Jan, Nina, and Morris.

At least two and not more than four employees must work each shift.
Edgar and Reuben cannot work together.
Morris will only work if Jan is also working.
Teresa and Reuben can work together only if Morris is working also.
If Jan is working, then either Morris or Reuben must be working.

Let's look at the summary we created for this game:

Employees: E, R, T, J, N, M

num. of emp. = 2, 3, or 4
If E, then no R, if R, then no E
If M, then J, if ^J, then ^M
T and R only if M
If J, then M or R, if ^M and ^R then ^J

5. Construct a diagram to incorporate all of the information in a general picture.
In order to answer the questions, you will need to be able to visualize the relationships between the various elements as defined by the conditions of the game. Diagrams are often the most effective way to organize all of the information presented to you in the scenario and conditions, and the diagram you create will enable you to quickly answer each question for the game. In the latter section of this chapter, we will discuss specific diagramming strategies for each of the major game types, but at this point we would like to give some general suggestions about the creation and use of diagrams.

If you are following this general strategy, you will not start creating the diagram until you have read through the scenario and the conditions so you are sure of what kind of game problem you are being asked to solve. Once you know what kind of game you are working on, you will have a better idea in mind of what kind of diagram will be most useful to solve the questions of the game. Also, keep in mind that some games, such as ones with many variable conditions and few fixed conditions, may not lend themselves to construction of a diagram until you have read the questions. If there are too many variables, and you think that the diagram will not helpful, wait until you have read the question and then decide. Indeed, some game questions can be answered directly from the summary of information that you create from the scenario and conditions.

Let's look at an example game for which a diagram is needed:

Sprinters from seven different countries are competing in the final of the 100 meter dash. Each country, Austria, Spain, Canada, Germany, Russia, the United States, and Morocco, has exactly one sprinter in the race. Participants have been assigned to lanes 1 through 8, with lane 1 on the inside (left) of the track, and lane 8 on the outside (right) on the track. One lane is left empty.

The sprinter from the United States is to the right of the sprinter from Canada.
The sprinter from Austria is in lane 2.
The sprinter from Morocco has the sprinter from Spain on one side of her, and the empty lane on her other side.
There is exactly one lane between the sprinters from Canada and Austria.

After reading through the scenario and the conditions, we can summarize first the roster of elements and then each of the conditions. You might come up with something like this:

sprinters: A, S, C, G, R, U, M

Lanes 1-8, one empty
C < U, U > C
A = 2
S M empty or empty M S
C _ A or A _ C

As soon as we read the scenario, we can see that this is an ordering game, in which we are asked to place elements in a line, from left to right. We can represent this graphically as:

Note: think about what the scenario is describing and then decide what kind of diagram would be most useful to represent the relationships among the different elements. If we need to place businesses on the floor of an office building, we would draw lines stacked vertically to represent the different floors. If we were asked to arrange people seated around a circular table, we would draw a circle.

Now, let's begin to fill in the information from our summary onto the diagram. You should always start with the conditions that are concrete, fixed. In this case, we know that A = 2, so we can place that sprinter in that lane.

What other conditions can we place on the diagram? Do we have any other conditions that are fixed? Well, the last condition tells us that there is exactly one lane between the sprinters from Austria and Canada. This sounds like a variable condition, and it is, until you combine it with the fact that we already know that the Austrian sprinter is in lane 2. If there is one lane between A and C, then C must be in lane 4. Note: If the condition had told us that A was in lane 3 instead of 2, we would not be able to place C with certainty. Instead, we would have to place C as being in either lane 1 or lane 5. Only because there is no lane that is two lanes from A on the left are we able to know that the sprinter from C must be on the right of A, in lane 4. Adding this new piece of information to our diagram, we now have something like this:

Now let's look at what other conditions we may be able to place on the diagram. We know that C < U, which would place U in either lanes 5, 6, 7, or 8. That might be helpful at some point, but we would just have to fill in the U with a little question mark for each of those lanes. Let's see if any of the other conditions can narrow down the possibilities further. We know that we must have S M empty or empty M S. This is a variable condition, however, if we look at our diagram and think about where we can place this set of three lanes, we can see that there are only two places where these lanes will fit, centered at lane 6 or centered at lane 7. Remember that we do not know the order of the three, whether it is S M empty or empty M S. Therefore, we have four possibilities:

At this point, it is easy to see that even though we have four possible arrangements, we can fit the U into each of these four diagrams into only two positions, either lane 8 or lane 5, in accordance with the condition that U is to the right of C. We now have a lot of information organized into a clear diagram that lets us easily see the possibilities.

Keep in mind that the basic strategy for making diagrams should be to place all the fixed conditions on the diagram first, then see if any of the variable conditions can be mapped. If there are multiple placement possibilities, it may be useful to draw out more than one diagram. However, if there are a large number of variable conditions, it may be better not to try to place all the possibilities on the diagram right away. Instead get the basic framework of the diagram down, and then go to the questions. Remember that questions will often add an additional condition, or modify one of the original conditions, and you can then place these new conditions on your diagram, but be careful to make sure you redraw the diagram with only the original conditions for the next question and do not carry over any of the additional information that was specific for only that question.

More detailed information on creating diagrams will be found in the latter portion of this chapter describing each of the four major game types.

6. Read the question; if any new information is provided, summarize that condition as well and add that to the existing diagram.
As we told you earlier, there are three kinds of questions. The first kind is answerable with only the information given in the scenario and the original set of conditions, the second provides a new condition, which must be considered when solving the question, and the third modifies or removes one of the original conditions. For these second two classes of questions, you need to summarize the new condition, and then add that information to your diagram.

Let's look at a sample question for the game we just set up:

Sprinters from seven different countries are competing in the final of the 100 meter-dash. Each country, Austria, Spain, Canada, Germany, Russia, the United States, and Morocco, has exactly one sprinter in the race. Participants have been assigned to lanes 1 through 8, with lane 1 on the inside (left) of the track, and lane 8 being on the outside (right) on the track. One lane is left empty.

The sprinter from the United States is to the right of the sprinter from Canada.
The sprinter from Austria is in lane 2.
The sprinter from Morocco has the sprinter from Spain on one side of her, and the empty lane on her other side.
There is exactly one lane between the sprinters from Canada and Austria.

1. If the sprinter from the United States is in the lane immediately next to the sprinter from Spain, what is the fewest number of sprinters that can be between the U.S. sprinter and the sprinter from Austria?

A. Zero
B. One
C. Two
D. Three
E. Four

First, let's look again at our set of four diagrams:

We now have a new condition stating that the U.S. sprinter is in the lane immediately next to that of the sprinter from Spain. This can be summarized as US or SU. If we look at our four diagrams, we see that only the first and the fourth placements will allow the U to be immediately next to the S. We can now modify our diagrams (only the first and fourth, since the other two placements will not allow the new condition to be met) as follows:

7. Use the diagram to answer the question.
It is critical that you read the question carefully. Pay very close attention to determine exactly what the question is asking. Even if you have followed the basic strategy up to this point, it is easy to misread the question and get the wrong answer. A question might ask which one of the following choices cannot be true, or which choice must always be true. These types of questions take a relatively long time since you need to then check each answer choice against your diagram to see which ones are possible, which are not possible, and which must be true.

Now when we go back to the sample question, it asks for the fewest number of sprinters that can be between the U.S. sprinter and the sprinter from Austria. If we look at our diagrams, beginning with the first one, there are five lanes in between the U and the A, though one of these (lane 5) is empty, so this would make four sprinters. If we look at the second diagram, where U is in lane 5, we see that there are only two sprinters in between them. Therefore the correct answer is choice C, two. Now, one thing that you need to be careful of is to not look at that second diagram and think that just because we have not assigned a sprinter to lane 3, there might not be a sprinter in that lane. If we consult the roster we created, we see that we still have not accounted for the two sprinters from Germany and Russia, and though we know that the two of them are in lanes 1 and 3, we do not know which sprinter is in which lane. The critical point here is to remember to consult the roster in order to remind yourself of all the elements in the game, especially the ones that are not yet fixed into a position in the diagram.

Again, here is the summary of the general strategy to use when approaching Logic Games:

3. Rewrite each condition on your scrap paper in symbolic form.

4. Check over what you have written to see if there is any additional information about relationships that can be deduced after considering the conditions together.

5. Construct a diagram to incorporate all of the information in a general picture.

6. Read the question; if any new information is provided, summarize that condition as well and add that to the existing diagram.

7. Use the diagram to answer the question.

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