Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is an Advanced Linear Game: Balanced, Defined.

This Defined, Balanced game is one of our favorites of all time. The game contains four variable sets: days of the week, morning activities, afternoon activities, and the four different activities. In setting up the game most students make a crucial mistake: they fail to show Thursday on the diagram. At first, this would not seem to be a big issue since the game scenario does not identify Thursday as a day on which Doctor Yamata works. But, because of the rule that states that “she lectures on exactly two consecutive calendar days,” the issue of consecutive days is critical, and a diagram without Thursday gives the false impression that Wednesday and Friday are consecutive. Once this mistake is made, the options for the LL block appear greater than they actually are, and making inferences becomes difficult. Fortunately, the first question of the game reveals that there is a major inference involving one of the lectures. Answering this question is critical to your success on the game. Remember, if you are faced with a Global question that indicates that one (or more) of the variables must be placed in a certain position (as with questions #8 and #12 of this game), you must answer the question. If you do not, you will miss a critical piece of information that will likely affect your performance on all other questions. In the case of question #8, if you do not have the answer when you arrive at the question, it is a fairly clear signal that you have missed something big in the setup.

When representing Thursday, mark each slot with an “X” in order to indicate that no work is done:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_1.png

Note that you could show Sunday but it too would have an X in both slots. With this basic diagram the rules can now be applied:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_2.png

Listing which activities occur in the morning and afternoon takes a considerable amount of time, and these activities have been placed on the right side of the diagram. The rules indicate that in the afternoon there will be three T’s and two L’s, and in the morning there will be three O’s, one R, and one T. This is extremely valuable information since it defines the composition of each row. Now that the rules have been added, we can begin to make inferences.

The first inference involves Saturday afternoon. According to the rules, Doctor Yamata can only lecture or treat patients in the afternoon. But on Saturday she cannot lecture, so it follows that she must treat patients, and a “T” can be placed on Saturday afternoon:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_3.png

Once T is established on Saturday afternoon, it becomes apparent that the LL block can only be placed on Monday-Tuesday or Tuesday-Wednesday:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_4.png

Note that Tuesday must always have a lecture and the other lecture will be placed on Monday or Wednesday, as shown by the split-option. Thus, in the afternoon, only two treatments remain to be assigned. One treatment will be placed on Friday afternoon (since it cannot be a lecture it must be a treatment), and the other treatment will fill in the Monday-Wednesday option:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_5.png

Consequently the afternoon spaces are filled, and it is revealed that only two solutions to the afternoon set exist:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_6.png

When seen in this light, it becomes obvious that the answer to question #8 is (B).

With the afternoon completed, we can now turn to an analysis of the morning row. Since Doctor Yamata cannot operate on Saturdays, on Saturday morning she is left with the choice of treating patients or conducting research. This is shown with a dual-option on Saturday morning:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_7.png

This leaves four morning spaces to be filled by three O’s and the remainder of the T/R dual-option. At first glance it may seem that no inferences can be drawn regarding the placement of these variables. However, the rule involving operations on Monday and Tuesday has a powerful effect on the possible placement of the three O’s: because only one operation can be performed on the Monday-Tuesday pair, this forces the other two operations to be performed on Wednesday and Friday:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_8.png

In summary, when an operation is performed on Monday, the operations rule prevents Doctor Yamata from operating on Tuesday; therefore, the remaining two operations must be performed on Wednesday and Friday. When an operation is performed on Tuesday, Doctor Yamata cannot operate on Monday, and the remaining two operations must again be performed on Wednesday and Friday. Hence, we can infer that operations are always performed on Wednesday and Friday.

The diagram is not yet complete. The final operation must be performed on Monday or Tuesday morning, next to the remainder of the T/R dual-option. This can be somewhat difficult to diagram, and we use a special parenthetical notation:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_9.png

The parentheses indicate that one of the two enclosed spaces must be an operation and the other space must be a treatment or research; it also indicates that the order is unknown. In this way the notation efficiently captures the four possibilities for Monday and Tuesday morning: OT, OR, TO, and RO. With this final piece the diagram for the game is complete:

Feb 93__M12_game#2_L11_explanations_game#2_setup_diagram_10.png

[Stephanie Oswalt]

We recently received the following question from a student. An instructor will respond below. Thanks!

Hi there,

I’m currently reading through your Logic Games Bible 2018 Edition. I am having trouble understanding the book’s interpretation of a given rule. Game #4: February 1993 Questions 8-12 (page 256) states as one of its rules that “If [Doctor Yamata] operates on Monday, she does not operate on Tuesday.” In the book’s analysis of this question set, the rule has been symbolized as a “Not-block” indicating that Operation cannot take place on Monday and Tuesday consecutively. Judging by the form of the rule “If - then” and this book's (and the Logical Reasoning book’s) suggestions, this rule should be diagrammed as a conditional, indicating that Om > ~ Ot or, If Dr. Yamata operates on Monday, she does not operate on Tuesday. Therefore, to say that If Dr. Yamata operates on Tuesday, she does not operate on Monday would be a Mistaken Reversal (to use the book’s terminology).

I would greatly appreciate your response (and clarification) for this question. I’ve been enjoying reading your guides and I believe I have understood all of the analyses up to this point. Please let me know if this is an error on my part (in which case a deeper explanation than provided would be very welcome) or if this has been addressed in future editions of the Bible.

Thank you for taking the time to field my question.

All best,

Megan

[Jonathan Evans]

Hi, Megan!

Thanks for the question! Actually, that's not a Mistaken Reversal; that's a valid contrapositive:

• OM OT

Flip the sides, then negate both sides. We end up with contrapositive:

• OT OM

This means that Doctor Yamata cannot operate on both Monday and Tuesday, from which we get the Not Block:

• [OM OT]

You are absolutely correct that this can be written as a conditional clue. The implications of this conditional and its contrapositive generate the Not Block above.

Excellent analysis. I hope this helps. Please follow up with further questions!

[officermeowmeow]

I just finished going through this logic game. I was confused by some of the wording and initially set up the original diagram wrong. I am hoping someone can explain the wording in the first rule.
Here's the game:
"Doctor Yamata....her work schedule must satisfy the following restrictions: She performs operations on exactly three mornings."

The next four rules make it clear that OOOTR (not in that order) will fill the morning slots and TTTLL (not in that order) will fill the afternoon slots.
I initially interpreted the rule as this: if she was not able to perform in the afternoon, the rule would have to be worded as 'She ONLY performs operations on exactly three mornings'. Since the original rule did not make it obvious (to me) that afternoon operations were off the table, I assumed she was still able to perform in the afternoon, as long as she performed operations for 3 mornings.
My worry is that, in the future, I will make the same assumption but the other rules won't be there to help fill in the other slots.
My question is, if this game only had the first rule and not any of the other rules, would Doctor Yamata be able to perform in the afternoon?
I appreciate any help/feedback! Thank you!

[Stephanie Oswalt]

Hi officermeowmeow (love the username )!

I have moved your post to the thread discussing this topic. Please review the above posts, and let us know if that helps, or if you still have further questions.

Thanks!

[silent7706]

Hi officermeowmeow (love the username )!

I have moved your post to the thread discussing this topic. Please review the above posts, and let us know if that helps, or if you still have further questions.

Thanks!

Hi Stephanie,

Can you please post the link where you move officiermeowmeow's question to? I'm struggling with the same issue actually and would like to understand how to avoid this confusion.

[Adam Tyson]

Hello, officermeowmeow and silent7706! The answer is that a rule like "she performs operations on exactly three mornings" does not, by itself, mean that she cannot also perform some operations in the afternoon. Saying "this must occur" establishes a minimum requirement, not an exclusive situation where nothing else can happen. It's only through the interaction with the other rules that we learn that she cannot operate in the afternoons.

Watch for words like "only" and "exactly" and "at least" and "not more than" to see if there are maximums or fixed amounts. When those ideas, which can also be expressed in other ways, are not present, then you should not assume that there is such a fixed "cap" on variables. Basically, if it isn't clearly ruled out, it's allowed!

[silent7706]

Hello, officermeowmeow and silent7706! The answer is that a rule like "she performs operations on exactly three mornings" does not, by itself, mean that she cannot also perform some operations in the afternoon. Saying "this must occur" establishes a minimum requirement, not an exclusive situation where nothing else can happen. It's only through the interaction with the other rules that we learn that she cannot operate in the afternoons.

Watch for words like "only" and "exactly" and "at least" and "not more than" to see if there are maximums or fixed amounts. When those ideas, which can also be expressed in other ways, are not present, then you should not assume that there is such a fixed "cap" on variables. Basically, if it isn't clearly ruled out, it's allowed!

Hi Adam,

Thank you for your response. I reread the rules and it became apparent to me that activities of for both morning and afternoon were essentially "fixed", since rules listed out five activities performed for morning and afternoon. Just another example that reminds me it's important to track numerical distribution!

[Anfernee9320]

Hi,

I have the same question as Megan from Post #2 why is the statement, "If she operates on Monday, she does not operate on Tuesday," not diagrammed as a conditional statement?

Number 6 of the conditional reasoning drill on page 76 of the current edition of the Logic Games Bible, presents a similar drill wherein, "If X arrives second, Y cannot arrive third."

My questions are;
1) What are the distinctions between the example on page 76 and the aforementioned conditional statement from the Dr. Yamata logic game?
2) Since the answer for the Dr. Yamata game is listed as a Not-Block, why is the game from page 76 not also listed as a Not-Block? The answer key on page 79 displays the answer as I would surmise: as a conditional statement.

Thank you in advance for the assistance.

[Adam Tyson]

The choice of the not block in this game can be thought of in part as a style choice, Anfernee9320, and it is a good option in this case because Monday and Tuesday are adjacent spaces in the linear diagram. The fact that you cannot have O next to another O in those two adjacent spaces makes the not block a visually powerful way to see that more clearly. The same style choice could have been made for that example in the LRB and would have been equally valid there, just as in this case you could have also chosen a conditional representation.

I would probably have chosen a double not arrow myself, just because I really like using those. That's this:

OM OTu

Either choice (not block or conditional) is valid and powerful! But imagine a hypothetical different rule that said "O cannot operate both Monday and Wednesday." In that case I would not use the not block and would recommend the conditional representation simply because the not block for that rule would not look like the setup of the game in the same way as it does in the original rule.

There can be many ways to represent a rule, and just because we show one way doesn't mean another way is incorrect! Study both of these options, and consider additional options, so that when you are faced with a situation like this on a test you can make the choice that works best for you.