- Sat Sep 20, 2014 11:00 pm
#40061
Setup and Rule Diagram Explanation
This is an Advanced Linear: Balanced game.
The game scenario establishes that each of five speakers must give a speech at one of two rooms—Gold and Rose. Additionally, each speech takes place at either 1, 2, or 3 PM. While each room must host one speech at 1 PM and one at 2 PM, it is unclear which room hosts the 3 PM speech. This uncertainty is troubling, but we can balance the game by adding a sixth variable (“E” for “Empty”) to designate the 3 PM slot in the room where no speech is being held.
This creates an Advanced Linear diagram, and because there are now six variables for the six positions, this is a Balanced game. With the basic structure in place, let us now turn to the rules.
The first rule establishes that M must be earlier than L, and in the same room as L. Your representation of this sequencing rule should also capture the grouping requirement that M and L be in the same room as each other:
This rule creates the following two Not Laws:
The second rule establishes that neither X or Y can be earlier than Z. The following sequence is impossible:
When handling a rule presented in the negative, immediately examine its positive implication: Z’s speech must be given earlier than X and Y, or else it must be given at the same time as either X or Y (Note that Z cannot be given at the same time as both X and Y, simply because at most two speeches can occur at any given time):
Since Z can be given at the same time as either X or Y, the second rule does not produce Not Laws for X or Y at 1 PM: it is entirely possible that either Z and X, or else Z and Y, are given at 1 PM. However, you should notice that Z cannot be given at 3 PM, because there is only one speech at 3 PM, and it is the last speech in the sequence:
The last rule presents a conditional statement: if L is in the Gold Room, then X and Z must both be in the Rose Room. You can either use “Gold” and “Rose” subscripts, or simply designate each room as “G” or “R”:
Since the necessary condition requires X and Z to be in the same room (the Rose Room), placing them in a horizontal block is more visually powerful. You should immediately make the connection with the previous rule, which ties X and Z into a sequence Z=====X. Of course, if X and Z must both be in the same room, they cannot be given at the same time. So, we can simplify the sequence as Z X:
You should also examine the contrapositive of the third rule: if either X or Z is not in the Rose Room, then L cannot be in the Gold Room. Thankfully, the dual-value system of the rooms allows us to represent the contrapositive as a positive statement: if either X or Z is in the Gold Room, then L must be in the Rose Room. And, since M and L must always be in the same room—with M earlier than L (first rule)—we can add that stipulation to the necessary condition of the contrapositive:
Note that L need not always be in a different room from X and Z! If L were in the Rose Room, for instance, the last rule would have no effect on the placement of X or Z (to conclude otherwise would be a Mistaken Negation). Likewise, if either X or Z were in the Rose Room, it would be possible for L to be in the Rose Room as well. Essentially, L cannot be in the Gold Room with either X or Z, but it can be in the Rose Room with either of them.
Thus, we arrive at the final setup for this game:
Before you proceed to the questions, it is important to recognize that the placement of Z represents a powerful point of restriction. This is because Z limits the placement of two other variables (X and Y), out of the remaining four. Since Z cannot be at 3 PM, it is a good idea to examine the next most restrictive hypothetical—placing Z at 2 PM.
If Z were at 2 PM (regardless of which room it is in), the second rule would prevent X and Y from being at 1 PM. Consequently, one of them would have to be at 2 PM, and the other—at 3 PM. However, this would force the remaining two variables—L and M—to both occupy the 1 PM slot, and in different rooms, which would violate the first rule. Consequently, Z cannot be at 2 PM, and must therefore be at 1 PM:
The inference that Z must be at 1 PM also suggests that L and Z can never be given at the same time:
This is an Advanced Linear: Balanced game.
The game scenario establishes that each of five speakers must give a speech at one of two rooms—Gold and Rose. Additionally, each speech takes place at either 1, 2, or 3 PM. While each room must host one speech at 1 PM and one at 2 PM, it is unclear which room hosts the 3 PM speech. This uncertainty is troubling, but we can balance the game by adding a sixth variable (“E” for “Empty”) to designate the 3 PM slot in the room where no speech is being held.
This creates an Advanced Linear diagram, and because there are now six variables for the six positions, this is a Balanced game. With the basic structure in place, let us now turn to the rules.
The first rule establishes that M must be earlier than L, and in the same room as L. Your representation of this sequencing rule should also capture the grouping requirement that M and L be in the same room as each other:
This rule creates the following two Not Laws:
The second rule establishes that neither X or Y can be earlier than Z. The following sequence is impossible:
When handling a rule presented in the negative, immediately examine its positive implication: Z’s speech must be given earlier than X and Y, or else it must be given at the same time as either X or Y (Note that Z cannot be given at the same time as both X and Y, simply because at most two speeches can occur at any given time):
Since Z can be given at the same time as either X or Y, the second rule does not produce Not Laws for X or Y at 1 PM: it is entirely possible that either Z and X, or else Z and Y, are given at 1 PM. However, you should notice that Z cannot be given at 3 PM, because there is only one speech at 3 PM, and it is the last speech in the sequence:
The last rule presents a conditional statement: if L is in the Gold Room, then X and Z must both be in the Rose Room. You can either use “Gold” and “Rose” subscripts, or simply designate each room as “G” or “R”:
Since the necessary condition requires X and Z to be in the same room (the Rose Room), placing them in a horizontal block is more visually powerful. You should immediately make the connection with the previous rule, which ties X and Z into a sequence Z=====X. Of course, if X and Z must both be in the same room, they cannot be given at the same time. So, we can simplify the sequence as Z X:
You should also examine the contrapositive of the third rule: if either X or Z is not in the Rose Room, then L cannot be in the Gold Room. Thankfully, the dual-value system of the rooms allows us to represent the contrapositive as a positive statement: if either X or Z is in the Gold Room, then L must be in the Rose Room. And, since M and L must always be in the same room—with M earlier than L (first rule)—we can add that stipulation to the necessary condition of the contrapositive:
Note that L need not always be in a different room from X and Z! If L were in the Rose Room, for instance, the last rule would have no effect on the placement of X or Z (to conclude otherwise would be a Mistaken Negation). Likewise, if either X or Z were in the Rose Room, it would be possible for L to be in the Rose Room as well. Essentially, L cannot be in the Gold Room with either X or Z, but it can be in the Rose Room with either of them.
Thus, we arrive at the final setup for this game:
Before you proceed to the questions, it is important to recognize that the placement of Z represents a powerful point of restriction. This is because Z limits the placement of two other variables (X and Y), out of the remaining four. Since Z cannot be at 3 PM, it is a good idea to examine the next most restrictive hypothetical—placing Z at 2 PM.
If Z were at 2 PM (regardless of which room it is in), the second rule would prevent X and Y from being at 1 PM. Consequently, one of them would have to be at 2 PM, and the other—at 3 PM. However, this would force the remaining two variables—L and M—to both occupy the 1 PM slot, and in different rooms, which would violate the first rule. Consequently, Z cannot be at 2 PM, and must therefore be at 1 PM:
The inference that Z must be at 1 PM also suggests that L and Z can never be given at the same time: