LSAT For Dummies, 2nd Edition (2014)
Part II. Analytical Reasoning: Following the Rules of the Logic Game
Chapter 5. Proper Placement: Analytical Reasoning Ordering Games
In This Chapter
Recognizing ordering games
Arranging the game board
Responding properly to ordering questions
Practicing on sample ordering games
At least one of the four logic games in any 35-minute LSAT analytical reasoning section requires you to order its elements. To make sure you're ready for this common question type, this chapter gives you tips on recognizing ordering games, dealing with their rules, and answering their questions correctly.
Spotting Ordering Games
Chapter 4 outlines the three steps to approaching logic games as read the facts, analyze the rules, and answer the questions. As you read the facts, your first job is to decide whether the game involves ordering or grouping.
The way you approach ordering games differs a bit from the way you handle grouping games, so distinguishing between the two is one of the first steps in your overall approach to the analytical reasoning section. (For the 411 on grouping games, see Chapter 6.) The LSAT is inconsistent in the number of ordering logic games it presents and the order in which they appear, so you have to rely on clues to know for certain whether you're dealing with an ordering problem.
Fact patterns and rules that deal with scheduling, ranking, or positioning are usually ordering games. You can spot them by looking for these word clues:
· Temporal references: Words such as day, year, and month indicate scheduling.
· Ranking terms: Comparison language — such as best/worst, greatest/least, and highest/lowest — show ranking.
· Positioning terms: References to positions — such as first/last and before/after — designate order.
Although the LSAT may throw in an unusual arrangement every once in a while, the most common ordering problems position elements (or game pieces) in a straight line from left to right.
If you're still not sure about the type of logic game after you read its setup, check the first question. The first question of each logic game usually asks you to choose a possible listing or assignment of the elements. First questions that ask for a possible listing, ordering, positioning, or ranking of the game pieces belong to ordering games.
Here's a fact pattern for a sample ordering game:
For its annual holiday concert, a high school orchestra performs exactly eight musical pieces — an aria, a concerto, a march, a polka, a rhapsody, a sonata, a tango, and a waltz — one immediately after the other. The conductor observes the following conditions in determining the order in which the orchestra plays the eight pieces for the concert:
This looks like an ordering game that involves positioning; it contains references to after and order in the facts.
To confirm your decision, see what you can glean from the first question:
· Which of the following could be the order, from first to last, in which the orchestra plays the pieces at the holiday concert?
(A) polka, march, sonata, aria, waltz, concerto, tango, rhapsody
The references to order, first, and last in the question tell you that this is most definitely an ordering game. The ordered list of eight musical pieces in the answer choice further confirms the classification.
After you know that you're working with an ordering game, immediately begin building your game board.
Becoming Chairman of the (Game) Board
Begin creating your game board as you read through the fact pattern. Putting a board together requires four steps: listing the game pieces, drawing the box chart, recording the rules, and analyzing them. You should spend only about four or five minutes completing this process. This section describes how to make the game board and analyze the rules for ordering games.
Putting together the game pieces
The first sentence usually tells you how many game pieces you're dealing with. Although ordering games commonly have six or seven game pieces, some may have fewer and some may have more.
The sample game we introduced in the preceding section has eight: “For its annual holiday concert, a high school orchestra performs exactly eight musical pieces… .” In the empty space in your test booklet, write out the game pieces using their first initials. For the sample game, you record A, C, M, P, R, S, T, and W to represent aria, concerto, march, polka, rhapsody, sonata, tango, and waltz.
Drawing the box chart
The box chart for most ordering problems is a simple table with as many columns as there are positions. The column headings for position and rank games are usually a number sequence, and the headings for scheduling games are time designations (such as hours in a day, days in a week, or months or quarters in a year).
The sample problem has eight positions designating the order in which the orchestra plays the eight musical works from first to last. Its column headings are simply 1, 2, 3, 4, 5, 6, 7, and 8, like this:
Draw only as many lines as you need to separate the elements in each column. On the LSAT, neatness doesn't count, but speed does.
Recording the rules
Most logic games have three, four, or five conditions or rules that restrict how you play with the pieces. For ordering games, the restrictions provide clues to how the pieces may be positioned in relation to one another. Common types of ordering rules are targets, spacers, and arrangers.
Targets give you concrete evidence about where a game piece belongs or doesn't belong on the board. You can record a target directly on the box chart.
Clues that tell you exactly where a piece belongs are rare on the LSAT. Here's an example:
The orchestra plays the march third.
You can record this target directly on the box chart like this:
More common on the LSAT are clues that narrow a piece's options on the board or that tell you where a piece doesn't fit. An example of the first type is this:
The third piece could be either the march or the polka.
You can indicate this restriction on the box chart by recording both options in the third column, separated by a slash to mean or:
Restrictions that tell you where a piece doesn't belong may be simply stated, like this:
The march is not played last.
You can record that clue by writing M in the last column and striking through it, like this:
Another way a rule may indicate that a piece doesn't belong is by giving you a few options where it may belong, like this:
The orchestra plays the sonata first, fourth, or sixth.
Record this clue on the box chart by showing the columns where S doesn't belong. You can keep track of where S can be by writing its options under the chart:
Some ordering rules indicate the number of spaces that separate certain pieces. Without additional information or analysis, you can't record these spacing rules directly on the chart, so remember them by recording them next to the chart.
When there are no spaces between pieces, they're ordered consecutively. A rule like this one tells you that one piece comes right after another:
The tango is played immediately after the rhapsody.
Record that the pieces are ordered consecutively by writing RT next to the box chart on your game board.
Read rules carefully. The rule mentions T first, but that doesn't mean that T comes before R and the order is TR. Because the rule states that T comes after R, the order is R before T, or RT.
Spacing rules may be precise or ambiguous. The preceding rule about the relative order of R and T is precise. You know that R is always immediately before T. A spacing rule may be worded more ambiguously, like this:
The tango is played either immediately before or immediately after the rhapsody.
You can record the two options next to the box chart on your game board like this:
RT or TR
Sometimes, a rule may tell you that pieces can't be next to each other:
The sonata is not played either immediately before or after the rhapsody.
We use strikethrough to indicate that S and R can't be consecutive, like this:
SR and RS
Spacing rules may tell you the exact number of spaces that separate certain pieces. A rule like this one indicates the precise number of spaces between two particular pieces but is ambiguous about the order:
The orchestra plays exactly two musical pieces between its performance of the march and its performance of the polka.
Use underscores to stand for the number of spaces between the playing of the two musical pieces:
M_ _P or P_ _M
The rule doesn't tell you whether M is before or after P, so either order is possible.
Rules that indicate the general arrangement of pieces without precisely stating how many spaces separate them are common on the LSAT. These rules tell you to order game pieces somewhere before or after other pieces. Here's an example:
The orchestra plays both the polka and the waltz after the concerto.
You know that C precedes P and W, but you don't know how many other pieces separate C from P and W. And you don't know the relative order of P and W. P could be played either before or after W, as long as both are played after C. We use a right arrow to designate arranging rules on the game board:
C → P
C → W
We use only right arrows for the arranger rule shortcut so that all the relationships go in the same left-to-right direction. So C → W is short for “C is before W” or “W is after C.”
Analyzing the rules
As you record each condition, consider how you may extend it to further develop your game board. You can better complete your board by expanding on the implications of individual rules and by combining rules that involve the same game pieces. Rules that offer contingencies may allow you to split the box chart and open up your ordering options. When you spend quality time extending your game board, you spend less time answering the questions.
One way to extend an ordering game board is by getting as much mileage from each rule or condition as you can. Spacing and arranging rules provide you with more information than just which piece comes before or after another. When you know the relative order of particular pieces, you can eliminate them from specific positions on the chart.
The sample rule that the tango is played immediately after the rhapsody not only indicates that R and T are consecutive but also that R can't be last and T can't be first. (If T is first, R can't precede T, and if R is last, there's no space for T to follow.) You can record this information directly on your chart:
The sample rule that C precedes both P and W reveals several additional conditions:
· C can't be last, because it precedes another game piece.
· C also can't be seventh, because it must have at least two spaces after it to accommodate both P and W.
· P can't be first.
· W can't be first.
You record these truths like this:
Because the rule doesn't give you the relative order of P and W, you can't know whether P precedes W or W precedes P. So you can't say that P or W can't be second. You only know that neither can be first because you have to leave a space for C.
Whenever an ordering game tells you the relative positions of its game pieces, you can eliminate each of these game pieces from either end of the order. Strike the piece that comes before another from the last position and the piece that comes after another from the first position.
Examining the rules in combination often provides you with valuable insight. As you record the rules on your game board, evaluate and combine the rules with common game pieces. For example, your game board could list the following conditions:
C → P
C → W
W → R
R → T
T → M
You know that C comes before both P and W, W comes before R, R is before T, and T precedes M. You can rewrite your rules in shorthand, like this: C → W → R → T → M.
P can be anywhere in the order as long as it's after C. So the order could be CPWRTM, CWPRTM, CWRPTM, CWRTPM, or CWRTMP. And because the sample game contains eight positions, the other two musical pieces could fit anywhere in between as long as one of these relative orderings is maintained.
You can use an expanded sequence to eliminate additional pieces from positions on the chart. For example, because C must be followed by W, R, T, M, and P, C can't occupy the last five positions on the chart. When you draw similar conclusions for W, R, T, and M, you can expand the chart like this:
Because you've eliminated four pieces from the first spot, you know that only A, C, or S can be played first.
When you start focusing your reasoning powers, you find that you can draw conclusions based on other spacing rules. Add this condition to the existing game:
Exactly three musical pieces separate the performance of the concerto from the performance of the sonata.
You can write this rule on the board like this: C _ _ _S or S _ _ _ C.
Additional consideration of the board reveals that S _ _ _ C isn't possible, because C can occupy only the first three positions. Therefore, S can be only fifth, sixth, or seventh. Your game board looks something like this:
That's a lot to work with, but there's still more. Notice that only two possible game pieces can occupy the first position: A or C. Knowing that either A or C can be first sets up a contingency, which may allow you to narrow the possible orders even further.
Working with contingencies
When you're left with two possible pieces for a particular position or when a given condition provides you with an either/or contingency, you may be able to split your board into more manageable possible orderings.
For example, you can build on the board you just created in the preceding section by splitting the chart into possible orderings when A is first and when C is first:
Placing C first puts S fifth and limits the second spot to P, W, or A and the last spot to P, A, or M. When A is first and C is second, S must be sixth, and only P or M can occupy the last spot. That's likely enough to get you going on the questions. You can consider the other possibilities when you start answering them.
Sometimes, a given condition sets up a contingency. You may see rules worded like this:
If the polka is played before the waltz, then the aria is played before the concerto.
Based on this rule, either of two possible rule pairings exists because either P comes before W or it comes after W. The first is a direct statement of the rule: P → W and A → C.
The other possibility, then, is W → P.
When you see such a contingency in the list of conditions for an ordering logic game, you likely can split your chart. In this case, you have one for when P → W (which also means that A → C) and one for when W → P.
Notice that applying this contingency to the game board you've developed for the orchestra game provides you with additional information to narrow the two existing options. When P is before W, you get two possible results:
· The first two rules can be combined as C → P → W → R → T → M.
· A must be before C, which places A in the first spot, C in the second, and S in the fifth.
When W is before P, you can come up with fewer deductions, but you can still create some possibilities that place P somewhere after W. In this case, W has to be second or third because W can never be first, and P, R, T, M, and S have to occupy the remaining spaces after W. You can record the possibilities on your chart like this:
These orderings show you that when P precedes W, you know the exact program schedule for the holiday concert.
Don't spend too much time coming up with all possible orderings. Get a handle on what's available given the two main options and draw additional conclusions as you answer the questions.
Answering Ordering Questions
When you've assembled your game board to the best of your abilities and within a reasonable time frame (about four to five minutes), you're ready to tackle the set of five to eight questions that follow the ordering game's facts and rules.
In Chapter 4, we discuss the four general logic game question types: possible listing/assignment, quantity, add-a-rule, and open. Less frequently you may see a couple of additional question types in one of the ordering games on your LSAT.
Substitute condition questions
Since 2009, a new question type has cropped up in the analytical reasoning section of the LSAT. It asks you to choose a condition in the answer choices that could replace one of the original conditions without changing anything about the orders. This new twist on analyzing conditions appears in some grouping games, but it's more common in ordering games. It's unlikely to appear more than once on your LSAT, and it's often the last question in the set, which is great because by the last question, you've established several possible orderings to draw from. Here's an example:
A circus performer includes exactly six activities in an act — bungee jumping, flame throwing, juggling, lion taming, sword swallowing, and trapeze swinging. The performer carries out each activity one at a time and no more than once during the act. The order in which the performer carries out the activities is subject to the following conditions:
· Lion taming occurs before bungee jumping.
· Flame throwing occurs immediately before bungee jumping.
· Trapeze swinging occurs before sword swallowing.
· Trapeze swinging occurs either immediately before or immediately after juggling.
Which one of the following, if substituted for the condition that trapeze swinging occurs before sword swallowing, would have the same effect on determining the order of the performer's act?
(A) Sword swallowing occurs before juggling.
(B) Sword swallowing occurs before flame throwing.
(C) Juggling occurs before sword swallowing.
(D) Sword swallowing occurs either immediately before or immediately after juggling.
(E) Sword swallowing may not occur first, second, or third.
To answer this question, create a game board:
1. List the game pieces: B, F, J, L, S, and T.
2. Create a box chart with six columns numbered 1 through 6.
3. Record the first rule: L → B.
4. Record the second rule: FB.
5. Record the third rule: T → S.
6. Record the fourth rule: TJ or JT.
7. Extend the board by entering on the chart where pieces don't fit and by combining rules, such as L → FB and TJ/JT → S.
8. Consider possible orderings.
The board looks something like this:
Here are the steps to answering questions that ask for a substitute condition:
1. Determine what effect the original condition has on the board.
It places T before S.
2. Check the answers to see which one would place T before S in all orderings.
3. Eliminate answers that don't jibe with your game board.
Choice (E) has to be wrong because S can be third in some orderings. Choice (A) puts S before T in all orderings, so it's out.
4. Rule out answers that don't have the same effect on the orderings as the original.
Choice (B) affects the order of S and F, but it doesn't put T before S in all orderings. You could have LSFBTJ, which doesn't maintain the third condition that T occurs before S. Choice (D) swaps S for T, which doesn't ensure that T occurs before S in all orderings.
5. Check the remaining answers to determine whether they affect the orders in the same way.
Choice (C) looks promising. Notice that whatever holds true for T is also true for J. The two acts, which are always consecutive, may hold the same positions, regardless of the orderings. Therefore, Choice (C) provides the same condition as the one that requires T to occur before S. T is interchangeable with J, so pick the answer that exchanges T for J in the original third condition. Choice (C) is correct.
Completely determined order questions
Another, even less common question is one that asks you to choose the condition that would allow you to create an exact ordering of the pieces. This question type may appear in an ordering set, such as the one we discuss in the earlier section “Working with contingencies.” The ordering game's conditions narrowed the possible orderings to these:
And you may be asked this question:
The order in which the orchestra plays the musical pieces at the holiday concert is completely determined if which one of the following is true?
(A) The polka is played second.
(B) The concerto is played first.
(C) The march is played last.
(D) The waltz is played fourth.
(E) The sonata is played last.
Attack this question methodically:
1. Eliminate obviously incorrect answers.
Choice (E) can't be right. The orchestra never plays the sonata last. Also notice that P can never be second, so Choice (A) is wrong. If P were second, C would have to be first, but that would put C before A and P before W, which violates the condition that A is before C when P is before W.
2. Dismiss conditions that could apply to more than one possible order.
Many of the possible orders have C in the first spot and M last, so Choices (B) and (C) are incorrect.
3. Check the remaining answer.
The last order on the chart is the only one with W in the fourth spot. When the waltz is fourth, all the other pieces fall into place. The four spots after W have to be filled by S, R, T, and M, so P and A have to join C in the first three spots. P comes before W, and when P is before W, A is before C. So A has to be first and C second. With C second, S has to be sixth, and R, T, and M fit into the fifth, seventh, and eighth spots, respectively, to retain their relative order. Choice (D) is the answer.
Ordering the Approach to an Advanced Game
LSAT ordering games are often closed, which means they involve an equal number of game pieces and positions to put them in. You use each game piece once and only once and fill each spot once and only once. However, not all ordering games are so, well, orderly. Some may use game pieces more than once; others may have fewer positions to fill than game pieces, meaning that you don't use all the pieces. Still others may require you to fill a position with more than one game piece. We refer to these more complex ordering games as open rather than closed. When you encounter an open ordering game, don't panic. You handle them in much the same manner as you do the more traditional variety. We show you how in this section with a sample practice problem.
A pastry chef must choose which four of five cake layers — chocolate, lemon, red velvet, strawberry, and vanilla — will make up the individual layers of a four-layer wedding cake. Each of the chosen layers will make up exactly one of the cake's four layers in order from bottom to top. The pastry chef is restricted by the following conditions in selecting the layers:
· If chocolate is included, then strawberry is the layer immediately above chocolate.
· Red velvet is neither the second layer nor the top layer.
· If vanilla is not one of the layers, then lemon is the second layer.
· If lemon is the second layer, then vanilla is not one of the layers.
You know this is an ordering game because the facts reference order and the conditions include ordering language, such as above, top, and second.
The five game pieces are C, L, R, S, and V, but your box chart has only four columns, one for each of the four layers represented from left to right in order from bottom to top. So you won't use one of the pieces when you put together your orders. Record the rules in shorthand:
· If C, CS.
· If R, R = 1 or 3.
· If no V, L = 2.
· If L = 2, no V.
Your board may start out looking like this:
If the open nature of this game unsettles you, gain your composure by working with the first question before you try to expand the conditions on your game board. Just run through the conditions and eliminate the answer choice(s) that violates each one.
Which one of the following could be the order of the layers from bottom to top?
(A) chocolate, strawberry, red velvet, lemon
(B) vanilla, lemon, chocolate, strawberry
(C) strawberry, red velvet, lemon, vanilla
(D) red velvet, lemon, strawberry, chocolate
(E) lemon, vanilla, red velvet, strawberry
To answer this first question, consider each condition. The first says “if C, CS.” Choice (D) places S before C, so it's out. The next states “if R, R = 1 or 3.” R is second in Choice (C), so it has to be wrong. For the third condition, eliminate answers that don't have V but put L somewhere other than second. Choice (A) has no V but puts L last, so it's out. Consider the fourth condition and look for answers that include V but put L in the 2 spot. Choice (B) does that, so it's incorrect. The only answer that doesn't violate a condition is Choice (E).
The first question of the set is almost always a possible listing/assignment question that doesn't require you to refer to a game board. So you can answer it before you create a board.
After you master the first question, return to your game board with renewed confidence. Examine the conditions again. Each one presents a contingency, and none allows you to add a permanent piece to the box chart. So try to come up with possible orders.
Notice that the third condition gives you a target clue in certain circumstances: If no V, L = 2. So create a possible layering for when there's no V.
When V isn't a layer, the four layers must be R, L, C, and S. L has to go in the 2 spot. C and S have to be 3 and 4 because they need to be next to each other in that order. The remaining first spot is for R. Only one order exists when V isn't a layer: RLCS.
Consider what happens when R isn't a layer and C, L, S, and V make up the cake. L can't be second, because V is a layer, and you need space for CS. Record these possible orders for when R isn't a layer on your game board: CSV/LV/L, LVCS, or V/L C S V/L.
What happens when C isn't a layer? L, R, S, and V must be layers. So L can't be in the 2 spot, and R must be 1 or 3. Write out the possibilities for when R is 1: RS/VLS/V or RS/VS/VL. When R is 3, you can have these orders: S/VS/VRL or LS/VRS/V.
Exclude L and work with C, R, S, and V. R has to be 1 or 3, and C and S have to be together. When R is 1, you can have these orders: RCSV or RVCS. When R is third, you have this possibility: CSRV.
Consider your options when S isn't a layer and C, L, R, and V are. Wait! That's not possible. You can't have C without S, so S has to be a layer in every cake.
Your expanded game board may look like this:
If you don't have time to come up with every one of these possible options, don't despair. At least get started on narrowing the possibilities so you get accustomed to the way you need to think to master the game.
Because you've spent quality time developing your game board, answering the questions shouldn't take much time at all. Consider this quantity question:
If the cake includes both a chocolate layer and a red velvet layer, then how many layers are there, any one of which could be the bottom layer?
Refer to your game board to see that when both C and R are cake layers, the only possible bottom layers are C and R. The answer must be two.
An add-a-rule question like this one is easy, too:
If the cake's bottom layer is vanilla, then which one of the following must be true?
(A) The cake doesn't have a chocolate layer.
(B) The cake doesn't have a red velvet layer.
(C) The cake's second layer is chocolate.
(D) The cake's strawberry layer is second.
(E) The cake's top layer is lemon.
According to your game board, the possible orderings when V is in the 1 spot are VSRL (if C is left out) and VCSL (if R isn't a layer). The only answer that's common to both options and therefore must be true is Choice (E). The top layer has to be lemon whenever the bottom layer is vanilla.
Try another question:
Which of the following CANNOT be true?
(A) The lemon layer is immediately below the red velvet layer.
(B) The red velvet layer is immediately below the chocolate layer.
(C) The red velvet layer is immediately below the strawberry layer.
(D) The strawberry layer is immediately below the lemon layer.
(E) The vanilla layer is immediately below the strawberry layer.
Compare the answer choices to your game board and eliminate all that could be or must be true. R is right before C when there's no L, so Choice (B) is out. R is right before S when there's no C, so Choice (C) is wrong. You see S right before L in one of the options when C isn't a layer, so Choice (D) isn't right. V comes right before S in the no C options, so Choice (E) is out. In no order does L come right before R, so the correct answer is Choice (A).
Other variations of the open ordering game exist, and you can handle them all when you apply what you know about working with the traditional types. To get more experience with the ordering logic game type, go to the practice exams in Chapters 15, 17, and 19 and online, purchase a copy of LSAT Logic Games For Dummies (Wiley), and order previous LSAT exam questions from the official LSAT website: www.lsac.org.