Month: August 2016

Logic Games: Must Be Easy

I have been struggling to explain “must be” questions to my logic games students. Sometimes they are super easy. Sometimes they are super hard. In the first case, you can read the correct answer right off your sketch. In the second case, the only way to solve one “must be true” question is to find four “could be false” solutions–and my students who are having trouble with logic games in general get tied up in knots trying to figure that out.

i have also been struggling with helping students know when to stop working out initial inferences. They know they are supposed to draw a sketch, write down the rules, and start looking for the obvious implications of those rules, but how do you know when you’re done?

As things would happen, it turns out that there is one simple answer to both of these difficult problems. I call it the “must be easy” rule.

Most logic game questions fall into one of two categories: “must be” and “could be” questions. “Which of the following must be true” is an obvious “must be” question, and so is “which of the following cannot be true.” In theory, the answer to “must be” question is something you might add to your logic game sketch. Thus, if question 3 asks, “Which of the following cannot occur on Tuesday,” you should be able to pencil in a “Not X” underneath Tuesday on your sketch.

A “must be easy” question is one where the answer is already penciled into your sketch. The only way to tell whether a “must be” question is a “must be easy” question is to look at it. It should take about three seconds per answer to decide whether you have already deduced the answer to this question. If you have, you’re done. That was fast–and easy!

If you haven’t already found the answer to your “must be” question, take another look at it. Is it a “focused” question? Logic game questions that ask about  a specific entity or slot are “focused,” a “must be focused” question is practically shouting at you, “Hey! There’s something about this slot or entity that you haven’t figured out yet.” If so, now is the time to think it through and add it to your sketch.

If it’s not easy, and it’s not focused, it may still be important–especially if it comes early in the game. I have seen this on several grouping games that have “/A->B” rules on grouping games.  (I have written about the importance of such rules here.) Take a moment to ask whether this is a “must be important” question.

If it isn’t easy, and it isn’t focused, and you can’t quickly think of any less-common but very-important deductions you should be making, it “must be hard.” That is a subject for another day. (Add hyperlink HERE when that day comes…)

Here are some examples from “10 Actual, Official LSAT Preptests Volume V.”

Preptest 62, Game 1, Question 4: if you already noticed that both gas and satellite must both fall on the last three days, Question 4 is a “must be easy” question. If not, now’s the time to ask, “What do I know about these days?”

Preptest 62, Game 2, Question 8: if you realized that “/P->O” means you have to have either purple or orange in every window, this “must be easy.” If not, this is neither easy nor focused–but it’s a critical inference. If you don’t understand that every window must have either purple or orange in it, this is one of the two hardest logic games in recent years. If you do see that the “/P->O” rule means “P or O” in every window, then it’s pretty straightforward.

Preptest 63, Game 1, Question 2: if you realized that “/H->P” means “H or P” on each court, including the appellate court which only has three slots and one is already filled, then this “must be easy.” If not, this is neither easy nor focused–but it is important! Question 3 and 4 both depend on exactly the same inference.

Preptest 63, Game 2, Question 7: if you noticed that W can’t come last and T comes before W so T can’t come next to last, then this was easy. If you notice that “at least two of the members dive after so-and-so,” then this is focused. The question is asking, “Who can’t dive last or next-to-last?”

Preptest 63, Game 3, Question 12: if you always knock out a whole staircase of entities at either end of a sequence that includes an ordered chain (“A…B…C” means B and C are out on the first day, and C is out on the second day, and A and B are out on the last day, and A is out on the next-to-last day), then this question “must be easy.” If you don’t, then it focuses your attention on Thursday.

No More Double-Nots

As I wrote recently, I have been trying to find a better way to explain the important features of a “/A->B” rule in a grouping game. The “/A->B” and “A->/B” rules can be literal “game changers” in any in/out, sorting, or matching game, but they are extraordinarily hard to explain. A student has to understand conditional reasoning thoroughly and have an almost intuitive grasp of how the necessary and sufficient terms work to be able to read either of those rules and see the implications in a grouping game. Most people who are that good at conditional logic don’t need me to tutor them.

PowerScore uses a special “double not arrow” (“<-|->”) to represent one of these rules, but I find this arrow hard to use and impossible to teach.  I have searched and searched for a better way to teach this concept, without any success, so finally decided I would make up new symbols for the “/A->B” and “A->/B” rules. Within a few hours of posting my idea, somebody suggested something even better (which proves that you can’t find everything you need to know by searching for it).

This helpful commenter reminded me that any conditional statement (“A->B”) can be written as an “or” statement (“/A or B”). This is something I teach all my students to help them understand why “unless,” “except,” “without,” and “until” all have a “not” in them. (“You won’t pass unless you understand logic can be written either as “/P or L” or as “P->L.”) That means that “/A->B” can be rewritten as “//A->B,” or, if you take out the double negative, “A or B.”

“A or B” is really easy to work with. In a logic game, I just write down A/B and I know exactly what to do with it. Each slot must have an A, or a B, or both. (You must always remember, of course, that “A or B” means “A or B or both.”)

What about the other rule, “A->/B”? The same rewriting approach gives us “/A or /B.” That works out to be the same as “NOT (A and B).” On a logic game, I write that as an [AB] block with a slash through it. I can’t have A and B together. If I’m working on an in/out game, that means at least one of those has to be “out,” so I can write “A/B” over in my “out” column.

If you can remember that these two rules produce “A/B” or “/[AB],” you’re already way ahead of most other test-takers. If you can remember which is which, you have a significant strategic advantage. Most grouping games with one of these rules in them have one or more questions that directly depend on them. I have seen grouping games with three questions that can be answered in 15 seconds or less if you start with “A/B” in the right slot.

So–how do you remember which is which? “/A->B” means that if you don’t have A, you must have B. That’s exactly what an “or” statement says. If you can remember that much, you can instantly rewrite “/A->B” as “A/B” every time. If you can’t you have to ask, “So what happens if A is ‘in’? I guess B can do whatever it wants. So  A and B can both be ‘in,’ so that means ‘A/B.'”

Better Than a Double-Not

I teach my students that as soon as they realize a logic game is a grouping game they should start tapping one foot and saying “count… count… count” in the back of their minds. Grouping games are all about counting. The instant you can fill up any group, you have usually solved the problem.

This priority on counting makes information about slots that have to be full (or empty) especially valuable. Two common conditional rules provide just that kind of information. They are:

  • A->/B
  • /A->B

The “/A->B” rule is so important for grouping games that PowerScore uses a special symbol (“A<-|->B”, or “the double-not arrow”) to note it. As a person who understands how important this is for grouping games, I think the “double-not arrow” is brilliant. As a tutor who wants to explain it to my students, I think it is both frustrating and confusing. That’s why I have come up with two arrows of my own.

Before I unveil my new arrows, let’s see why it is so important to spot the “double-not arrow” situation and so confusing to use it. Let’s walk through what “/A->B” really means. In a grouping game, we tend to think of items as “in” or “out” rather than “true” or “false,” so we’ll use in/out terminology for this discussion.

  • If A is OUT, B is IN means:
    • Either A or B must be IN
    • A and B cannot both be OUT
    • A and B can both be IN

This means that any in/out game with a “/A->B” rule will always have either A or B “in.” By the same logic, and in/out game with the other rule (“A->/B”) will always have either A or B “out.” That is essential information! But how do you teach that to a student who is still trying to figure out the basics of conditional reasoning?

I have searched the Internet looking for clues. This is a PowerScore symbol, so I figured they must have a way to teach it. If they do, they haven’t printed it or posted it yet. Instead, they have forum discussions where they try to help people untangle themselves after they get it all confused–which is what I have been doing. Up until now.

What we need here are some simple symbols that make this easy and obvious. Fortunately, not only can we come up with such symbols, we can write them out with a keyboard. Note how the slash comes first in the “/A->B” situation, but comes second in the “A->/B” case. Let’s turn those slashes into pictures. If we put the forward slash first, we can make a “/\” picture. If we put it second, we get a “\/” picture.

  • /A->B turns into A<-/\->B
  • A->/B turns into A<-\/->B

Pictures are helpful if they mean something, so let’s call the “/\” picture an “erupting volcano.” The “erupting volcano arrow” means that something is erupting, so that something must be in your slot. The “\/” looks like a “leaky funnel,” which means something is leaking, which means something must be out.

If you can remember that “slash comes first” means “/\,” and “/\” means “erupting volcano,” and “erupting volcano” means something must be in, you can turn a “/A->B” rule into a full slot within seconds. And if you can remember what a “leaky funnel” does, you’ll fill an out slot just as fast.

And… if you’re tapping your foot, saying, “count… count… count” in the back of your  head, that full or empty slot just made the game much easier!