Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Pattern Game.

Some games are difficult because they contain a large number of rules or variables; other games are difficult because they are built around a complex concept that forces you to make several deep inferences. This game is one of the latter. And as is always the case with these type of games, creating an effective setup makes the game much easier. The first important decision in this game is how to display the dancing couples for each dance. Many students attempt to use a basic setup similar to the following:

o93_game_#3_setup_diagram 1.png

Unfortunately, this setup does not offer you a concrete base to work with. In each of the three dances you are still making decisions about each of the six variables, eighteen total for all three dances. If at
all possible, you would like a setup to fix some of the variables and thus allow you to reduce the number of variables you have to work with for each dance. The following basic setup achieves that goal:

o93_game_#3_setup_diagram 2.png

By fixing R, S, and T, only the three boys remain to be placed for each of the three dances. This effectively reduces the total number of variables to be placed to nine, down from eighteen in the previous setup. Choosing to display R, S, and T is superior to choosing K, L, and M because it allows us to display the second rule within the diagram:

o93_game_#3_setup_diagram 3.png

Note that this second rule involves a double arrow. If we know who partners Rita in dance 2, that person must partner Sarah in dance 3, and if we know who partners Sarah in dance 3, that same person must partner Rita in dance 2.

According to the last rule, “No two children can partner each other in more than one dance.” By combining this rule with the second rule, we can make the essential inference that the boy who partners Rita in dance 2 and Sarah in dance 3 must partner Tura in dance 1. This pattern holds true for every possible game configuration. This inference can also be added to the setup:

o93_game_#3_setup_diagram 4.png

Since the game now has one established pattern, it is quite possible that other patterns exist. For example, let’s examine the boy who dances with Rita in dance 1. In dance 2 this same boy can partner Sarah or Tura, and in dance 3 he can partner only Tura (remember, Sarah is already taken and he can’t partner with Rita again). But wait—if he must partner Tura in dance 3, then he cannot partner her in dance 2, and thus he must partner Sarah in dance 2. It therefore follows that whomever partners Rita in dance 1 must partner Sarah in dance 2 and Tura in dance 3. Now that this second pattern exists, there can be only one possible pattern for the boy who partners Sarah in dance 1: Tura in dance 2 and Rita in dance 3 (remember each of the other two girls in dances 2 and 3 are involved in other patterns). Thus, by analyzing the interaction of the second and third rules we have established the three patterns that must exist in every game:

• One boy must partner T in dance 1, R in dance 2, and S in dance 3. (T - R - S)
  One boy must partner R in dance 1, S in dance 2, and T in dance 3. (R - S - T)
  One boy must partner S in dance 1, T in dance 2, and R in dance 3. (S - T - R)

Obviously, uncovering this pattern within the grouping rules will now allow us to easily conquer the
game. But, as we begin the questions, we also need to keep in mind that Karl must partner with Sarah
in dance 1 or 2 (essentially this means that he will be in either the STR pattern or the RST pattern).

This game was universally considered by students to be the hardest game on the October 1993 LSAT, if not one of the hardest games of the modern era. However, an application of the basic rules allows any student to answer at least the first four questions, and those students who discovered the three patterns found the game quite easy. Remember, just because a game contains a few simple rules doesn’t necessarily mean that the setup is also simple or uninformative. Always examine the interaction of the rules, even if there are only two or three.

[desmail]

Hi,

I am having extreme difficulty understanding Game 2 on page 309 in the Logic Games Bible (the recital game).

The second rule states: "Whoever partners Rita in dance 2 must partner Sarah in dance 3."

This rule does not state that if for example Luis partners Sarah in dance 3, then he is automatically partnered with Rita for dance 2. That would be a mistaken reversal. The rule only states that if 2, then 3.

However, in the explanations (pg 310), they say that this rule should be represented with a double arrow, and that it applies both ways. How can this be true? I did not make this inference and as a result had so much difficulty completing the game.

I understand why this is so, because if K is put in dance 3 with sarah and not in 2 with rita, then the person assigned to 2 with rita cannot be assigned to 3 with sarah because that place would have been filled already by K. But what I don't understand is how I would have caught that inference without looking at the answer.

Am I missing something? Is the rule worded in a way where I am supposed to assume that this is a double arrow representation? Because from reading it initially, it looks like it is a conditional statement with a single arrow. I am so confused!

Thank you,
Dana

[Dave Killoran]

Hi Dana,

Thanks for the question. You've run across an interesting situation, one where additional information forces the arrow to go both ways. So, let's discuss exactly what's going on.

As you note, the rule as stated is: R2 S3. By itself, there is no reversal of this rule, and there is nothing in the language that indicates that S3 R2 is present. So, your direct interpretation of the rule is correct. But, there's more going on in this game, and that other information forces the rule to reverse.

The easiest way to understand what occurs is to realize that every girl has a dance partner for each dance, so Rita always has a dance partner in dance 2. Of course, if we know who partners with Rita in dance 2, then we automatically know who partners Sarah in dance 3. That comes directly from the conditional language in the rule. But, consider for a moment the scenario that occurs when we know who partners Sarah in dance 3. Let's say it is Luis who partners with Sarah in dance 3, and we know this must be true. Who then, can partner with Rita in dance 2? Could it be Miguel? No, because then Sarah would have to partner with Miguel in dance 3, and we know that has not occurred because Sarah is already with Luis. Could it be Karl that partners with Rita in dance 2? No, again because then Sarah would have to partner with Karl in dance 3, and we know that has not occurred because Sarah is already with Luis. Thus, when Sarah is known to partner with Luis in dance 3, Rita has no other choice than to partner with Luis in dance 2. That effect results in a the double arrow you see on the diagram.

So, why does this happen? It happens because of the unusual language in the rule stating that the two females partner the same boy in the two dances. In essence, this creates a block effect, and once you know who one of the girls partners with, you know who the other partners with.

Another way of thinking about this is to take the contrapositive of the rule while keeping in mind that they partner with the same boy in the two dances. Here's the contrapositive:

Not S3 Not R2

So, if Sarah doesn't partner with a certain boy in dance 3, Rita cannot partner with that boy in dance 2. So, if Sarah is already known to partner with a certain boy in dance 3 (say, again, Luis), that eliminates every other boy from partnering with Rita in dance 2 (meaning Miguel and Karl are out, leaving only Luis available). That results in a double arrow as well.

It's definitely tricky, and you ask a great question. Please let me know if that explanation makes sense. Thanks!

[desmail]

Thank you so much, I really appreciate the help! Your explanation made perfect sense.

[Dave Killoran]

Thanks--glad I could help!

[AylixW]

Hi,

I tried doing the game as a timed challenge as the homework suggested, but had no idea how to start with the setup. I read the explanation immediately after the game, but am having a hard time understanding the set up. I don't know if there is an easier way to describe it or maybe someone could try explain it in different words?

Thank you!!!

[Dave Killoran]

Hi Aylix,

Thanks for the question. That's a pretty lengthy, involved explanation in the book, so to re-create that in a different direction without being totally sure of where you ran into trouble would be pretty difficult

When you say that you had trouble understanding the setup, can you be more specific? Was it the choice of variables for the base? The rules? The inferences? Any help you can give me would allow me to address that element in more detail.

All that said, this is a very unique game, and certainly not the easiest game out there.

Thanks!

[AylixW]

Hi Dave,

I just read through the explanation again carefully and now fully understand how to diagram and answer the questions very easily. I am just a little nervous about being able to do a similar game on my own as I would never have thought to use R S and T as fixed variables and then go on to see the three patterns: T-R-S, R-S-T, and S-T-R. I guess this will just take practice, hopefully a similar game won't be the October 6th LSAT!

Thanks so much!!!

[reop6780]

On the stimuli, "whoever partners with Rita in dance 2 must partner Sarah in dance 3"
I say ?R(2) _> ?S(3)

I would like to know whether the reverse of it is also right.
?S(3)_>?R(2)

What allows the reverse of the original condition right if it is correct?

(I just solved it allowing the reverse to be right without certainty and at least the answers are correct. However I'm aware of dangers of assuming things especially with conditional reasoning.)

[Dave Killoran]

Hi Hyun,

In this case, that apparent reversal works because of the way the partners are matched with each other. Because the same partner must be involved in both dances, once you know who that person is in one of the named dances, you automatically know it is the same partner in the other dance.

Thanks!