Setup and Rule Diagrams

[Administrator]

This is a fairly straightforward Grouping game, where seven candidates--Q through X--are available to fill just three spots, with one of those three spots assigned the title of Leader. Because the numbers are given to us this is a Defined game, and with seven variables to fill only three spaces it is Unbalanced: Overloaded. Fortunately that seeming imbalance can be quickly resolved by showing four spaces in your "out" group, something like this:

__ __ __ | __ __ __ __

L

Now we have seven spaces for our seven people, and all we have to track is the in/out nature of our rules and inferences. Note too how I've designated a spot with an L for Leader. Just be careful not to confuse that with a Not Law and you're good (although that would be tough to do here since there are no L variables).

The three rules are also relatively benign, where the first rule establishes that if Q or R is in then it's in the L position (this means that Q and R can never both be in--there's only one L after all--so one of your out spots can be filled with a Q/R); rule 2 is a simply S --> T; and rule 3 is a slightly more complex W --> R+ V.

The contrapositives of those last two rules are crucial here! For rule 2 consider what happens when T is out: S is out as well, and now three of your out spaces have been filled by Q/R, T, and S. That leaves only one more open space out before the game gets very, very restricted! And when we think of what we're told by rule 3, that W and V cannot be selected together, then you know an out spot must be filled by the W/V split, as well.

And this leads to one of the most powerful inferences in the game: when T is out, then all four of your unselected spaces have been filled: Q/R, T, S, and W/V. That means your "in" group is: Q/R as Leader, W/V (avoiding R and W together), and our wildcard variable X. By filling the out group we de facto fill our in group, with X guaranteed and the rest extremely limited.

This singular notion immediately answers question 4, very nearly solves 5, and goes a long way towards answering 3. It's Grouping 101 that you always pay close attention to both the selected and unselected sets, and this shows why.

Another interesting point to make about this game is that opportunities abound to consider, or even completely reuse, prior work. This is a huge bonus to students who don't erase earlier diagrams and who remember exactly what they've already done as they move through the questions. Take question 3: what could allow V to be the Leader? The correct answer to 1 has V as the leader, so should immediately be considered (and sure enough, spotting that Q and R need to be out so neither is the Leader, and that T needs to then be in accords nicely with answer choice C in the first question).

[gwlsathelp]

This is a general question about the phrasing of rule #3: "if Wells is a project member, neither Ruiz nor Verma can be". I took "nor" to mean the negative of "or" and wrote the rule as...

W R or V

Is "nor" always "and"?

[Jeremy Press]

Hi gw,

The short answer to your question is yes. The phrasing "neither [X] nor [Y]" should always be interpreted, in all contexts (conditional or otherwise), as "not [X] and not [Y]."

Here's another example from a different (non-conditional) context: "Alice can finish the race neither [immediately before] nor [immediately after] Bob." That would mean Alice finishes the race "not [immediately before]" Bob AND "not [immediately after]" Bob. So we'd have two not-blocks.

I hope this helps!

[Duval]

Hello,

I understand that W and V can never be together, but R can also never be present with W. My question is why you chose to represent rule 3 with W V instead of W R. With way, in my setup, I did not separate the relationship between the three variables and thus missed the W/V split taking up a spot in the out group.

[KelseyWoods]

Hi Duval!

You are correct that rule 3 can be represented as W V and W R. W can be with neither V nor R. The reason that we only show that either W/V must be in the out group is that R is already in a similar out group relationship: the first rule tells us that Q and R cannot both be in because if either of them is in, they have to be the leader and there's only one leader. In general, if you have a variable that cannot be with multiple other variables, you can only show one of those relationships in the out group slots. So we can have Q/R and W/V in the out group. But if I try to add another out group slot for W/R, that doesn't quite work, because both W and R could already be in those other out slots. Essentially, those dual options in the out group are saving out group slots and helping you figure out at a minimum who you need room for in the out group. Between Q, R, W, & V, we always have to have at least 2 of those 4 out. So at a minimum we need to save two out group slots for those four variables. But if I did Q/R, W/V, and W/R, that would make it seem like they needed to take up at least 3 out slots. Which, again, is not true because if I had both W and R out, then V and Q could both be in.

So if you have any rules about 2 variables that cannot be together, you should save one of the out group slots for one of those variables UNLESS one of those variables is already sharing an out group slot with another variable that it cannot be with.

Hope this helps!

Best,
Kelsey

[anabel.masaschi@gmail.com]

I am a bit confused about the Q/R split. The way the rule is phrased, does that mean that ONLY Q or R can occupy the leader space? How did you discern that they couldn't both be in the out group?

[Adam Tyson]

Happy to clear that up for you, Anabel! That rule does not require Q or R to be the leader, and they can both be out. All that rule requires is that IF one of them is in, THEN that one is the leader, and we can infer that the other one could not also be in. That leaves us with three broad possibilities regarding those two variables:

1: Q is the leader, R is out. We cannot infer anything else about the group.
2: R is the leader, Q is out. The rules require that W is also out.
3. Someone other than Q or R is the leader, so they are both out. Combining that with the fact that at least one of W and V must also be out, we can infer that T must be in, because if T was out, S would also be out, and we would be unable to find three variables that could all be in together.