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#41098
Setup and Rule Diagram Explanation

This is a Partially Defined, Balanced, Identify the Templates, Grouping game.

The game requires us to assign six variables (rangers) to three different groups (areas). Since each variable is used exactly once, the game is Balanced. However, the game is only Partially Defined, because the number of variables in each group can vary from one to three.

With all the rules represented, your initial setup should look like this:
PT79_Game_#2_setup_diagram 1.png
That last rule requires some additional analysis. You can see above that it gives a pair of double-arrow relationships, where the first half of the rule creates the O2 arrow to the JK block, and the second half of the rule creates an arrow in the reverse direction (J and K together means O must be in 2). It effectively functions as an if but only if statement: “O is in area 2 if but only if J and K are assigned to the same area.” From that we can take a double-arrow contrapositive, where the absence of either condition (O not in 2, or J and K not together) tells us the other cannot happen.

At first glance, it may be tempting to delve into a deep analysis of the Numerical Distributions governing the assignment of rangers to areas. After all, there are only two distributions in play: 3-2-1 and 2-2-2. However, these distributions are largely unfixed because none of the rules limit the precise number of rangers assigned to each area, and this fact tends to limit their utility somewhat. (There is no rule telling us, for instance, that more rangers are assigned to area 2 than to area 1, or that exactly twice as many rangers are assigned to area 3 as to area 2).

More specifically, unfixed distributions are always worth noting, and depending on the game can prove quite significant, but as we will see they do little for us here. Why? Primarily because the three areas are not functionally identical: almost all the rules seek to limit the rangers’ assignment to particular groups, rather than distribute the rangers into functionally identical groups. Thus, a distribution of 3-2-1 would be much more useful if we knew precisely which groups have 3, 2, or 1 rangers assigned to them, i.e. if the distributions were fixed. So the numbers matter, but our ability to assign sizes to individual groups is extremely limited in this game and that restricts their overall impact.

A better way to proceed is to combine the rules in the hopes of making a few inferences. For instance, since L must be assigned to the same area as either K or M, but not to the same area as both, it is clear that K and M can never be assigned to same area as each other. Thus, K cannot be assigned to area 3.

And as noted previously the last rule deserves our attention: not only is it a complicated rule and thus risky to treat dismissively, but it is also one that “splits” the solutions to this game into two distinctly different directions: either O is assigned to area 2, or it is not. Let’s examine them separately:

Template 1: O is assigned to area 2.
PT79_Game_#2_setup_diagram 2.png
In this template, K and J must be assigned to the same area as each other. Since K is never assigned to area 3 (see above), J cannot be assigned to area 3 either. So, it seems that the JK block must be assigned to either area 1 or area 2, however this is only partially correct. If J and K are both assigned to area 2 (along with O), then area 2 would be maxed out. In accordance with the second rule limiting the placement of L to the same group as either K or M, we need to assign L to area 3. Unfortunately, this leaves P to be assigned to area 1, in violation of the second rule. Since at least one ranger must be assigned to each area, this solution is impossible. Therefore, in Template 1 the JK block must be assigned to area 1:

Template 1: O is assigned to area 2.
PT79_Game_#2_setup_diagram 3.png

The alternative scenario created by the last rule is to not assign O to area 2. In that case, we need to ensure that J and K are not assigned to the same area as each other. Furthermore, since O is never assigned to area 1 (second rule), this scenario requires that O be assigned to area 3 (along with M).

Template 2: O is not assigned to area 2.
PT79_Game_#2_setup_diagram 4.png
We could continue building upon this template by using the LK block or the LM block, which would require splitting the solution off in three different directions (the LK block can be assigned to either area 1 or area 2, whereas the LM block can only be assigned to area 3). This approach could yield some benefit, but the risks (the time required as well as the potential for leaving a possibility out) outweigh the benefits for all but the most advanced test takers. It is better to develop a comfortable relationship with degrees of uncertainty, and move on to the questions!
 amiru77
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#39623
Hi PS!
I was wondering if a template approach would be helpful for G2 Lsat Sep 2016? I tried G1 with template and it took me longer overall time to do this game with template rather than without? I think this is the case for the second one two?
thanks,
Amir
 James Finch
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#39647
Hi Amiru,

Looking at this game and its rules, I would definitely take the approach of setting up four templates:

One with M-L-O in area 3 together; one with M-O in area 3 together, but not L; one with M-L in area 3 together, but not O; and one with M in area 3 but neither L nor O.

Three of the four templates have very little variability, and even the fourth deals with a block of three variables. So a template approach works nicely for this game.
 andbzav@gmail.com
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#67984
Hi,

I'm having trouble understanding the double arrow created by the last rule.

I diagrammed the first part of the rule as O2 -> JK and then created a separate conditional for the latter half of the rule, namely not O2 -> not JK.

Seeking further clarity as to how the second half of the rule creates an arrow in the reverse direction, from JK to O2.

Thank you
 andbzav@gmail.com
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#67987
I think I figured it out, but validation if correct would be much appreciated!

The rule is saying that both O2 and JK are sufficient AND necessary for the other to occur? I got this much by reasoning as follows:

First part of the rule: O2 -> JK. Second part of the rule: not O2 -> not JK.

The contrapositives of the first and second parts of the rule are what make O2 and JK both sufficient and necessary? I hope that makes sense. Breaking down the rule into these "parts" is the only way I could make sense of it although I recognize its not at all efficient to see it this way.

Thank you
 Jeremy Press
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#67991
Hi andbzav,

You've got it exactly right! It's the contrapositives of each part of the rule that give you the ability to run the arrow in both directions. One rule of thumb applicable to rules like this: if you see a two-part conditional rule in which the second part of the rule looks like the "Mistaken Negation" of the first part of the rule (note, it's not a true Mistaken Negation, because it is a rule!), you can use a double arrow diagram for both parts.

I hope this helps!

Jeremy
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 tabook
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#86088
Just to clarify, from my understanding is that you created that second template of o not being in area 2 by using the contrapositive of the rule. Is that something you recommend for other games ? using the contrapositive of a rule ( depending on the rule ) to create a second template ?

thank you
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 KelseyWoods
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#86127
Hi Tabook!

The decision about when to use templates is highly dependent on the full context of the game and how all of the rules and game concepts interact with one another. We would never really say "anytime you have this one type of rule, you should do templates" because it depends on what the other rules are and how they are affected by the rule you're considering basing templates around.

In terms of doing a template based on the contrapositive of a rule, I would say that's not something you would typically do but it can be appropriate/useful in certain situations.

A typical contrapositive pair doesn't actually set up only 2 possibilities. For example, if I have the following contrapositive pair:
A :arrow: B
B :arrow: A

There's really 3 possible combinations of A and B that work: I could have both A and B, I could have neither A nor B, or I could have just B but not A. So I can't just do one template for if I have A and one template for if I don't have B. I'd also have to do a template where I have B but not A. Is making this many templates worth it? Only if there are several other variables that are impacted by whether I have A, not B, or B but not A.

But in this case, we have a bi-conditional. In a bi-conditional, there really are only 2 options. We either have O in area 2 and J & K are together, or we have O not in area 2 and J & K are not together. There is no 3rd option here. So a bi-conditional has fewer options than a regular conditional statement. That still doesn't necessarily mean that you automatically do templates every time you have a bi-conditional. But if there are other variables that are impacted by the bi-conditional, it can be a useful strategy.

Hope this helps!

Best,
Kelsey

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