Setup and Rule Diagrams

[Jon Denning]

After two very reasonable, Linear-centric games (an Advanced and a Basic), we find ourselves faced with even more: another Basic Linear! At this point the closest thing we've seen to a group this whole section was the photo essays being assigned themes in Game 1...a conspicuous and unusual absence, but if you're a Linear fan nothing to be sad about.

Game 3 has six obstacles—which I'll abbreviate as R, S, T, V, W, and Z—being ordered into a sequence of six positions, 1-6. So not only is this more Basic Linear, but like Game 2 it's a perfectly-balanced, 1:1 distribution. We're on the LSAT bunny slopes here.

Under most contexts (i.e. not LSAT Logic Games) you'd be right to call this "boring" so far. Certainly it's safe to label it "familiar."

We have just three rules here:

S is either 3 or 4, which can be shown above your 1-6 base as a split option with S/ and /S above spaces 3 and 4.

W is "just before" Z, which is a WZ block. That's an odd phrasing, "just before," but all it means is that they're
consecutive. This block creates some immediate Not Laws: W not 6, Z not 1. When combined with the first rule
about S it also creates two more Not Laws: the WZ block can't take both of S's spots, meaning W cannot be 3,
and Z cannot be 4 (interestingly this never gets tested directly in this game, but it's always a good idea to
recognize how a variable that requires a position or two, like S here, affects a block that's sliding around, like WZ).

R and V are not consecutive, in either order. A double-not block, with RV and VR. This doesn't tell
us anything just yet, but it does mean we'll need to be sure we leave room for these two to separate (allowing
at least one thing to get between them).

It's not a rule, but I'd also note that T is our lone floater.

This is the point in the game where I'd check the clock and likely make the prudent decision to dive into the questions, knowing full well that another inference or two might still be possible (there may be more to show in our currently-spartan diagram, but blindly searching for potential, non-obvious inferences at this stage is wasteful).

It just so happens, in fact, that there IS another nice inference in the form of a variable (T) restriction, but it would be rather remarkable for you to have noticed it up front. Fortunately, and is very often the case when it comes to obscure but universal truths in LG, a question eventually reveals it to you: question 14 asks you to list all of the places T could go, meaning that's a variable to look more closely at to see if you can find an issue. Would you have known to suspect T of causing problems in your setup? No way! But now we should be suspicious.

And sure enough, what you find is that if T is in 3 or 4, it forces S into the other spot (4 or 3), and that creates two regions of open space—spots 1 and 2, and spots 5 and 6—thereby creating two blocks from the remaining variables: the WZ block from the second rule which now goes in either 1-2 or 5-6, and an RV/VR block in the other open two-spot space. Of course R and V in a block breaks the rule that they can't touch, so there's your T-issue: T cannot be in 3 or 4!

Put another way, what would force R and V into a block and thus cause problems? Well we already have a WZ block, so if S and T are ALSO a block in the very middle then our six variables would be in three pairs: WZ (always), S and T, and R and V (bad). So we can never allow S and T to form a center block themselves, meaning T cannot be next to S by going in 3 or 4.

Great inference, even belatedly

Alright, back to it.

So while I'd be deep into the questions at this point, hammering away and luxuriating in my saved time, another (admittedly far-fetched) option does exist: Templates.

Yes, as we saw in the first two games, the possibility to create Templates is present here! They're less apparent, and as we'll see from the sheer number we need to make less tempting (hopefully) than in Games 1 and 2, but since we can do it I'll list them out for anyone who's curious.

First though, what alerts me to the possibility? A few things, turns out. One, and most immediately, is that S in either 3 or 4 creates two paths, and along those paths the WZ block could potentially be quite restricted. For instance, I can see clearly that if S is 3 and WZ goes 1-2, the final three spots 4-6 have to be R, T, V (with R and V split up, in some order, separated by T in 5). Ditto if S is 4 and WZ are 5-6: R, T, V up front in 1-3, with separation involved (and the R/V order unknown).

So in certain instances the Templates look really limited, and thus very attractive!

But the key to Templates is that you account for ALL possible outcomes, not just the most restricted or well-determined ones, so we'd also need to think about what happens when, say, S is 3 and the WZ block comes after it, somewhere down in 4-6. That's far less singular of an occurrence, and that wider range of possibilities is what would absolutely dissuade me from the Template approach in this game.

Like I said though, for the sake of comprehensiveness let me list out what you could show if you chose to show everything:

Template 1: S 3, WZ 1 and 2 (2 Solutions)

W Z S R/V T V/R [note that T must be in 5 to keep R and V apart here]

Template 2: S 3, WZ 4 and 5 (4 Solutions)

(T, R/V) S W Z V/R [more movement here; T and either R or V are in 1-2 in some order, with the other of R/V in 6]

Template 3: S 3, WZ 5 and 6 (4 Solutions)

(T, R/V) S V/R W Z [like Template 2; T and either R or V are in 1-2 in some order, with the other of R/V in 4]

Template 4: S 4, WZ 5 and 6 (2 Solutions)

R/V T V/R S W Z [here T must be in 2 to keep R and V separate; similar to Template 1]

Template 5: S 4, WZ 1 and 2 (4 Solutions)

W Z R/V S (T, V/R) [again, less certain; T and either R or V go in 5-6, with the other of R/V in 3]

Template 6: S 4, WZ 2 and 3 (4 Solutions)

R/V W Z S (T, V/R) [like Template 5; T and either R or V go in 5-6, with the other of R/V in 1]


So that's 6 Templates with a total of 20 different solutions. Not the worst thing I've ever seen, but almost certainly more than you'd want to commit to up front (especially in a Basic Linear game with only a 5 question reward).

Instead what I'd be conscious of is that the placement of S has a big impact on the WZ block, and vice versa, and the need to keep R and V apart requires some careful planning as well.


Last thing: because I'm nearly certain I would neither use Templates myself for this game, nor recommend them for others, in explaining the questions I'm not going to refer to Templates or act as though I'd have them handy for solving.

[Etsevdos]

What are your thoughts on breaking into two templates with S in 3 / 4 and noting the not laws in either instance? I found that helpful for me and a good middle approach.

[Claire Horan]

Hi Etsevdos,

In my own approach to the problem, I stopped just where Jon recommends, and went on to the questions. The reason is that once you find the aforementioned not-laws, it will be easy to deal with the rules when you get to questions addressing them. But since drawing two templates with S in either 3 or 4 takes barely any time, I'd say there's no problem with doing that if you prefer. The inference about the floater T not being able to go 3rd or 4th is easier to see once you have the two templates, so if it worked for you, I would say it's a fine approach!

[SGD2021]

For a game like this, can we assume that none of the variables repeat because of the following language: the obstacle course “will consist of” six separate obstacles? (So since it’s telling us the course will consist of the 6 obstacles, all six must show up and since there are only 6 possible slots, no repeats are possible?)

[Adam Tyson]

I wouldn't call that an assumption, SGD2021 , but yes, that language proves that each obstacle must be used exactly once. Otherwise the course would not consist of six separate obstacles.