linear games Flashcards
concept of linearity
linearity involves the fixed positioning and ordering of variables
in every linear game, one of the variable sets is chosen as the base and is diagrammed in a straight line, either horizontally or vertically, and the remaining variable sets are placed into slots above or next to the base
variable sets with the greatest sense of inherent order almost always create the best base, because they provide a logical framework within which to place all other variable sets
one-to-one relationships
in a one-to-one variable set relationship, each variable fills exactly one slot and there are the same number of slots as variables to be placed
two primary considerations when representing the rules
- how to diagram the rule itself
- how to show the implications of the rule on your main diagram
four main types of linear rules
- variable placement rules
- fixed position rules
- sequencing or relative placement rules
- conditional rules
variable placement rules
variable placement rules specify where a variable or variables must be placed or cannot be placed
in representing the rules, you should always search for what must be true and what cannot be true; by defining these endpoints, you can define the range of possibilities within a game
not laws: physically notate where a variable cannot be placed
dual options: when only two variables that can be placed in a single slot
split-dual options: when a single variable has only two possible positions
triple options: when a single space can contain only one of three variables
fixed position rules
fixed position rules specify where a variable must be placed or cannot be placed in relation to another variable
blocks
in linear games, blocks reflect the idea of a fixed spatial relationship between variables
blocks represent variables that are next to one another, not next to one another, or separated by a fixed number of spaces; the most basic block indicates that two variables are adjoining
split blocks
a split block indicates that the variables are separated by a fixed number of spaces
can play a powerful role within certain games
frequently-used language constructions:
- spaces ahead or spaces before
- spaces behind or spaces after
- spaces between or spaces separated by
rotating blocks
here, the order of the variables isn’t given
write out both possibilities when diagramming these sort of rules
not blocks
aka negative blocks
indicate that variables cannot be next to one another, or cannot be separated by a fixed amount of space
should be diagrammed with a slash between the two variables
sequencing rules
sequencing rules establish the relative positioning of variables
whereas block rules precisely fix the variables in relation to each other, sequencing rules don’t
to represent sequential relationships, draw a straight line between the two variables; this shows that it’s a relative indicator, and the position of the two variables isn’t fixed
branched sequences
can be double- or triple-branched
show that some variables (which may or may not have relationships with each other) all occur before or after another variable
conditional rules
this is the most complex linear game rule
conditional rules appear most frequently in grouping games
conditional reasoning is a fundamental component of LR and LG
conditional reasoning involves sufficient and necessary conditions, and are often formed using the if… then construction
when a sufficient condition occurs, the necessary condition must occur; but when the necessary condition occurs, the sufficient condition only may occur
when diagramming conditional statements, use an arrow construction; the sufficient condition goes before the arrow, and the necessary condition goes after
terms to introduce a sufficient condition
if
when
whenever
every
all
any
people who
in order to
terms to introduce a necessary condition
then
only
only if
must
required/precondition
unless
except
until
without
unless equation
in the case of “unless” (and its synonyms “except,” “until,” and “without”), a special two-step process called the Unless Equation is applied to the diagram
- whatever term is modified by “unless” becomes the necessary condition
- the remaining term is negated and becomes the sufficient condition
inferences
inferences are relationships that must be true in a game, but are not explicitly stated by the rules or game scenario
one of the keys to powerful logic game performance is making inferences after you have diagrammed all of the rules
in some games, a single inference can be the difference between the game seeming easy or difficult
inference-making strategies
- linkage
simplest and most basic way to make inferences
involves finding a variable that appears in at least two rules and then combining those two rules
often, the combination will produce an inference of value - rule combinations
classic combinations that always yield certain inferences
doesn’t rely on using a connecting variable, but rather using known variable placements to infer remaining placements - restrictions
always look to the restricted points—the areas in the game where only a few options exist—for inferences
if you can identify a restriction, generally there are inferences that will follow from your examination of that point
the trick is to determine exactly where the restrictions in a game actually occur
false inferences
- conditional rule reversal
assuming that when the necessary condition occurs, the sufficient condition must also occur
aka mistaken reversal - misinterpreting block language
“before” and “after” mean different things than “between” - false blocks
- false not-block inferences
balanced vs unbalanced games
balanced games = number of supplied variables equals the number of available slots, resulting in a one-to-one relationship of variables to slots
unbalanced games = either have fewer variables than available slots (underfunded), or have a greater number of variables than available slots (overloaded)
unbalanced games tend to be more difficult than balanced games, and overloaded scenarios often the hardest type of linear games
numerical distribution
allocates one set of variables among another set of variables
numerical distributions occur in every type of game except mapping games
logical opposite
vs
polar opposite
logical opposite = any statement that contradicts the statement in question. literally, anything different from the statement
polar opposite = a statement that contradicts the statement in question as completely as possible
falsity conversions
must be false —> cannot be true
not necessarily true —> could be true
could be false —> not necessarily true
cannot be false —> must be true
six specific question types
- list questions
- maximum/minimum questions
- “5 if” questions
- justify questions
- suspension questions
- rule substitution questions
list questions
list questions present a list of variables that can either fill an individual slot or possibly solve the game
two basic types of list questions:
complete solution list question
complete variable list question
complete solution list question
this type of list question asks you to select the answer choice that contains a solution to the game that doesn’t violate any of the rules
the best technique for solving this type of list question is to take a single rule and apply it to each of the five answer choices, one at a time
if you run out of rules, see if the game has any limiting features you otherwise overlooked
to solve these questions, don’t choose a random game rule to apply. instead,:
- choose rules that are simple and visually easy to apply
- begin by choosing rules that are as concrete as possible, and have as few options as possible
- choose rules that invoke the game’s number rule, if any
complete variable list question
this type of list question supplies a list of all the variables that could be placed in a single slot, and asks you to select the answer choice that contains all the variables that could occupy a certain spot of any game scenario
to solve these questions,:
- first check the rules to see if any variables cause a violation when placed in the slot
- second check all Not Laws
- finally, make hypothetical solutions to prove or disprove any remaining answers
maximum/minimum questions
maximum/minimum questions generally ask you to identify the greatest or least number of possibilities in a certain scenario
two types:
- questions that ask for the maximum or minimum number of possible variables that can fill a certain position, or the maximum or minimum number of positions a variable can fill
- questions that ask for the maximum or minimum number of variables that can be placed between two particular variables, or ask for the maximum or minimum number of spaces from one variable to another
in order to answer these questions, you must control the variables in order to either maximize or minimize the situation
also consider the following steps:
- check prior work to evaluate answer choices
- consider the abstract relationships created by the rules
- if multiple answer choices remain, attempt to create a solution that matches the greatest/least possible number remaining among the answers
in evaluating these questions, consider the following points:
- the answer choices often feature numbers as the answer
- the questions are often global but can be local
- although they include ‘could’ in the question stem, they are more often than not ‘must be true’ questions
- each question has both a maximum and minimum component, whether or not it’s being tested
“5 if” questions
these questions are identifiable not by the question stem, but by the language of the answer choices
each answer choice begins with the word “if,” making each choice a whole new scenario (and essentially whole new question)
these questions are incredibly time consuming; consider leaving them for last
justify questions
justify questions ask you to select the answer choice that forces a specified result
here, the answer choices create the effects, and the correct answer is the effect that’s specified in the question stem
to solve a justify question,:
- first, carefully analyze the result that’s specified in the question stem (which is what you’re working backwards from). this process typically centers on which variables are the most powerful, and which are the most restricted
- second, analyze the answer choices to determine which is most likely to create the type of limitations that will result in the desired outcome
one recurrent type of incorrect answer choice has been an answer that could create the desired outcome, but doesn’t have to create that outcome
suspension questions
suspension questions always appear at the end of the game, and they suspend one of the game rules
these tend to be time consuming, since you have to redraw the diagram and re-evaluate the relationship of the rules
removing a rule typically changes the game’s inferences, and usually creates a greater number of solutions than originally existed
rule substitution questions
rule substitution questions involve suspending one of the game’s original rules, and ask you to select an answer choice that contains language that has an identical effect as the original rule
aka, you must find an answer that presents a perfect equivalent of the original rule
rule substitution questions are an unusual combination of suspension and justify questions
these are a relatively new type of lsat question, and they always appear at the end of a section
question solution strategies
solve for the correct answer
(global questions: refer to the inferences in the game setup; local question: consider the effects of the condition in the question stem)
process of elimination
(you don’t have to prove the right answer as correct; you can also prove the wrong answers as incorrect)
hypotheticals
(a self-created solution that can help eliminate or confirm certain answer choices
re-using information
(the information gleaned from answering some questions aid in answering other questions)
placing diagrams
(write diagrams neatly and in big, empty spaces; for local questions, write mini-diagrams next to the relevant questions)
diagramming local questions
flawed approaches:
- write on the main diagram
- create a diagram grid
dead vs. active rules
dead rule = one that either has been fully encompassed within your diagram or that no longer needs to be actively considered
active rule = must still be considered when solving questions; applies to the majority of game rules
