- Wed Jun 29, 2016 10:02 am
#26787
The second game presents a Defined Grouping scenario (also, once again Unbalanced: Overloaded, as we'll see), where a history project has four students out of a group of six candidates assigned to one of four years: 1921-1924. This is clearly an Unbalanced situation (more people than spots), but the six students--L through Y--can, much like we saw in the first game, be balanced by including an out group:
__ __ __ __ | __ __
1 2 3 4
With this simple setup we've also introduced the Linear aspect of the game, since the 1-4 years (I'm abbreviating 1921-1924 with their units digits) of the assigned group are sequential, and at least one of the rules includes the concept of ordered placement (rule 4, as we'll see).
What is striking to me about this setup, and hopefully got your attention as well, is how small the out group is: only two members! The significance of that really can't be overstated. It means that as soon as we can determine just two people who are unassigned, the entirety of the project's membership is known. And sure enough, several of the questions revolve around that very idea.
The rules are primarily Grouping in nature, and present the following:
Space 3 must be either L or T. Note that this doesn't mean L or T can't go somewhere else (both L and T can both be used, so long as one is on 3). This only prohibits the other four students from going in that space.
M can only go in 1 or 2. Again, pay close attention to a third option for M: out. It's only when M is assigned to the project that she's in the first two years. I showed this simply as a Not Law for M under 4 (M not 3 is implied by the first rule), and a conditional: M --> 1 or 2.
If T is assigned then so is R. This is also conditional: T --> R. But the key, and it's one of the most powerful ideas in this game, is the contrapositive, No R --> No T. Think about what happens when R is out: T must also be out, which fills our two out spots and forces the other four people--L, M, O, and Y--to be in. That would put L on 3, M in 1 or 2, and O/Y in 4. By removing R an entire cascade of consequences occurs, and if you're focused on the heavily restricted unassigned pair you'll catch it.
Lastly, and combining both conditionality and linearity, the fourth rule tells us that if we have R then we get an OR block (in that order). Three things stand out with this rule: first, R becomes very limited in its placement options. For instance, R cannot go in 1 because then there would be no room for O ahead of it. Similarly, R cannot go in 4 because that would force O into 3, which must be either L or T. That means that if R is assigned, it MUST be in 2! That's the only position for R where O could immediately precede it. (Careful here: that doesn't mean that O is similarly limited; O is only tied to this rule when R is in, so until that happens O is free to do as he pleases)
Second, since R can only go in 2, forcing O into 1, then M is left with nowhere to go. Remember, M had to be 1 or 2, so if we fill those spots with the OR block then M is out. Take away? M and R can never be in together; one of the two must always be out! But can they both be out? No! Don't forget that if R is gone then T is gone too, and that fills our out group. So R or M in = the other out. R or M out = the other in.
The third notable aspect of rule 4 is its connection to rule 3, where T --> R. The shared R between these rules is important: T tells us R, which tells us the OR block is in 1 and 2. So if L is out and T is 3, then we'd have OR in 1-2. If T is 4, then L is 3 and OR is 1-2. But what if T is 1, or 2? Then we have a problem: O and R need those spaces if T is in. Inference? T cannot be in 1 or 2.
You're probably getting the sense at this point that there are a lot of Not Laws present in this game. You're right. Take a look:
That's a lot of useful information, and it makes the questions entirely manageable.
__ __ __ __ | __ __
1 2 3 4
With this simple setup we've also introduced the Linear aspect of the game, since the 1-4 years (I'm abbreviating 1921-1924 with their units digits) of the assigned group are sequential, and at least one of the rules includes the concept of ordered placement (rule 4, as we'll see).
What is striking to me about this setup, and hopefully got your attention as well, is how small the out group is: only two members! The significance of that really can't be overstated. It means that as soon as we can determine just two people who are unassigned, the entirety of the project's membership is known. And sure enough, several of the questions revolve around that very idea.
The rules are primarily Grouping in nature, and present the following:
Space 3 must be either L or T. Note that this doesn't mean L or T can't go somewhere else (both L and T can both be used, so long as one is on 3). This only prohibits the other four students from going in that space.
M can only go in 1 or 2. Again, pay close attention to a third option for M: out. It's only when M is assigned to the project that she's in the first two years. I showed this simply as a Not Law for M under 4 (M not 3 is implied by the first rule), and a conditional: M --> 1 or 2.
If T is assigned then so is R. This is also conditional: T --> R. But the key, and it's one of the most powerful ideas in this game, is the contrapositive, No R --> No T. Think about what happens when R is out: T must also be out, which fills our two out spots and forces the other four people--L, M, O, and Y--to be in. That would put L on 3, M in 1 or 2, and O/Y in 4. By removing R an entire cascade of consequences occurs, and if you're focused on the heavily restricted unassigned pair you'll catch it.
Lastly, and combining both conditionality and linearity, the fourth rule tells us that if we have R then we get an OR block (in that order). Three things stand out with this rule: first, R becomes very limited in its placement options. For instance, R cannot go in 1 because then there would be no room for O ahead of it. Similarly, R cannot go in 4 because that would force O into 3, which must be either L or T. That means that if R is assigned, it MUST be in 2! That's the only position for R where O could immediately precede it. (Careful here: that doesn't mean that O is similarly limited; O is only tied to this rule when R is in, so until that happens O is free to do as he pleases)
Second, since R can only go in 2, forcing O into 1, then M is left with nowhere to go. Remember, M had to be 1 or 2, so if we fill those spots with the OR block then M is out. Take away? M and R can never be in together; one of the two must always be out! But can they both be out? No! Don't forget that if R is gone then T is gone too, and that fills our out group. So R or M in = the other out. R or M out = the other in.
The third notable aspect of rule 4 is its connection to rule 3, where T --> R. The shared R between these rules is important: T tells us R, which tells us the OR block is in 1 and 2. So if L is out and T is 3, then we'd have OR in 1-2. If T is 4, then L is 3 and OR is 1-2. But what if T is 1, or 2? Then we have a problem: O and R need those spaces if T is in. Inference? T cannot be in 1 or 2.
You're probably getting the sense at this point that there are a lot of Not Laws present in this game. You're right. Take a look:
That's a lot of useful information, and it makes the questions entirely manageable.
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