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- Mon Jun 25, 2012 8:49 am
#88283
Setup and Rule Diagram Explanation
This is an Advanced Linear: Balanced game.
In this game seven suspects are each questioned once, on consecutive days with one suspect questioned each day. This configuration creates a Basic Linear game that is Balanced. However, the end of the game scenario introduces another variable set—the confessions—that connects to each suspect, and turns this into an Advanced Linear game. The confessions variable set is easy to miss if you read too quickly. There are different ways to represent the confessions, but we will use “C” for confess and “NC” for not confess, leading to the following basic setup:
With this basic setup in mind, we can analyze each of the rules.
Rule #1. This rule is easy to represent by placing T on the third day:
Of course, with T removed from consideration, there are now only six suspects to consider.
Rule #2. This rule is also easy to represent, with a NC on the fourth day:
Rule #3. This rule creates a W
S sequence, which then yields two Not Laws:
Rule #4. This rule creates another sequence, which can be represented as:
This sequence yields Not Laws showing X and V cannot be questioned first, and Z cannot be questioned on the last two days:
Rule #5. This rule establishes that any suspect questioned after W does not confess. By itself, this rule can be diagrammed as:
There are other ways to diagram this rule as well, but none is preferable to another.
By combining this rule with the third rule, we can deduce that S does not confess:
By itself, this rule is somewhat powerful. However, this rule will become considerably more useful once we examine the sixth rule.
Rule #6. This rule establishes that exactly two suspects confessed after T was questioned. From the first rule we know that T was questioned on day three. Thus, on exactly two days out of days four through seven the suspects confessed, and on the other two days out of days four through seven the suspects did not confess. Rule #2 already established that on day four the suspect did not confess, which means that out of days five, six, and seven, suspects confessed on two days and did not confess on the other day.
The information above is powerful, but do not stop linking the rules just yet. From the combination of the third and fifth rules we know that S does not confess and that S is questioned after W. Because no suspects confess after W is questioned, and two suspects must confess within the grouping of days five, six, and seven, we can infer that S must be questioned last, and that W must be questioned on day six. Of course, because S is questioned last, S does not confess. With S not confessing, we can infer that the suspects questioned on days five and six do confess.
This powerful information leads to the following setup:
While the information above is significant, the placement of W and S leads to new inferences when the fourth rule is revisited. Specifically, only four variables remain unplaced: Y, V, X, and Z. Y is random, and thus can be questioned on any of the four remaining open days. V, X, and Z are all involved in the fourth rule, and from that rule we can deduce that Z cannot be questioned on day four or five, the last two remaining open days (otherwise there would be no room to question X and V). Thus, Z must be questioned on day one or two. Further, since X and V cannot be questioned on day one, only Y or Z can be questioned on day one, leading to a Y/Z dual-option on day one.
When the above information is added to the prior setup, we arrive at the final setup for this game:
While arriving at this setup takes time and work, now that we have this setup and all of the information it contains, the questions should not be difficult. To work with this setup most efficiently, focus on the four unplaced variables— Y, V, X, and Z—and on the confession status of the suspects questioned on the first three days.
This is an Advanced Linear: Balanced game.
In this game seven suspects are each questioned once, on consecutive days with one suspect questioned each day. This configuration creates a Basic Linear game that is Balanced. However, the end of the game scenario introduces another variable set—the confessions—that connects to each suspect, and turns this into an Advanced Linear game. The confessions variable set is easy to miss if you read too quickly. There are different ways to represent the confessions, but we will use “C” for confess and “NC” for not confess, leading to the following basic setup:
With this basic setup in mind, we can analyze each of the rules.
Rule #1. This rule is easy to represent by placing T on the third day:
Of course, with T removed from consideration, there are now only six suspects to consider.
Rule #2. This rule is also easy to represent, with a NC on the fourth day:
Rule #3. This rule creates a W
S sequence, which then yields two Not Laws:Rule #4. This rule creates another sequence, which can be represented as:
This sequence yields Not Laws showing X and V cannot be questioned first, and Z cannot be questioned on the last two days:
Rule #5. This rule establishes that any suspect questioned after W does not confess. By itself, this rule can be diagrammed as:
There are other ways to diagram this rule as well, but none is preferable to another.
By combining this rule with the third rule, we can deduce that S does not confess:
By itself, this rule is somewhat powerful. However, this rule will become considerably more useful once we examine the sixth rule.
Rule #6. This rule establishes that exactly two suspects confessed after T was questioned. From the first rule we know that T was questioned on day three. Thus, on exactly two days out of days four through seven the suspects confessed, and on the other two days out of days four through seven the suspects did not confess. Rule #2 already established that on day four the suspect did not confess, which means that out of days five, six, and seven, suspects confessed on two days and did not confess on the other day.
The information above is powerful, but do not stop linking the rules just yet. From the combination of the third and fifth rules we know that S does not confess and that S is questioned after W. Because no suspects confess after W is questioned, and two suspects must confess within the grouping of days five, six, and seven, we can infer that S must be questioned last, and that W must be questioned on day six. Of course, because S is questioned last, S does not confess. With S not confessing, we can infer that the suspects questioned on days five and six do confess.
This powerful information leads to the following setup:
While the information above is significant, the placement of W and S leads to new inferences when the fourth rule is revisited. Specifically, only four variables remain unplaced: Y, V, X, and Z. Y is random, and thus can be questioned on any of the four remaining open days. V, X, and Z are all involved in the fourth rule, and from that rule we can deduce that Z cannot be questioned on day four or five, the last two remaining open days (otherwise there would be no room to question X and V). Thus, Z must be questioned on day one or two. Further, since X and V cannot be questioned on day one, only Y or Z can be questioned on day one, leading to a Y/Z dual-option on day one.
When the above information is added to the prior setup, we arrive at the final setup for this game:
While arriving at this setup takes time and work, now that we have this setup and all of the information it contains, the questions should not be difficult. To work with this setup most efficiently, focus on the four unplaced variables— Y, V, X, and Z—and on the confession status of the suspects questioned on the first three days.
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Dave Killoran
PowerScore Test Preparation
Follow me on X/Twitter at http://twitter.com/DaveKilloran
My LSAT Articles: http://blog.powerscore.com/lsat/author/dave-killoran
PowerScore Podcast: http://www.powerscore.com/lsat/podcast/
PowerScore Test Preparation
Follow me on X/Twitter at http://twitter.com/DaveKilloran
My LSAT Articles: http://blog.powerscore.com/lsat/author/dave-killoran
PowerScore Podcast: http://www.powerscore.com/lsat/podcast/
