# If O is summarized, which one of the following CANNOT be true?

Natalle on August 2, 2019

Set up

Hi, I originally skipped this game since it seemed like it had an added complexity with which game would be summarized and those that would not. But it does have a linear order as well. How would you set up your diagram? Thank you.

Irina on August 2, 2019

@natallekezy,

Great question. This fairly challenging game requires us to determine the order of the books as well as which books are summarized. Let's look at the setup:

6 books - FKNORT are discussed in six weeks:

___ ___ ___ ___ ___ ___
1 2 3 4 5 6

Written summaries are required for one or more books, but we are not given any information on how many exactly, only that it has to be at least 1.

Let's look at the rules:

(1) No two books that are summarized are discussed in consecutive weeks.

~ss

(2) If N is not summarized, then both R&T are summarized

~ N -> R ^ T

and logically equivalent statement is:

~R v ~ T -> N

Note that we cannot infer anything from R & T being summarized, it is entirely plausible that R T N are all summarized.

(3) N is discussed earlier than T & T earlier than O

N > T > O

Let's see what inferences we can make for our order:

___ ___ ___ ___ ___ ___
1 2 3 4 5 6
~O ~O ~N ~N
~T ~T

(4) F is discussed earlier than O and O is discussed earlier than both K &R

F>O> K & R

Combined with rule (3), we can conclude that:

N > T
> O > K &R
F >

We can see that there are at least 3 books before O and 2 afterward, meaning O must be book #4.

N/F O K/R K/R
___ ___ ___ ___ ___ ___
1 2 3 4 5 6
~O ~O ~K ~N ~N
~T ~K ~R ~F ~T
~K ~R ~F
~R

K & R must go into weeks #5 & #6 in some order, #1 must be either N or F, #2 could be N/ F/ T and #3 could be T/F depending on which book goes first:

Option 1 -N is #1:

N T/F T/F O K/R K/R
___ ___ ___ ___ ___ ___
1 2 3 4 5 6

Option 2 - F is #1:

F N T O K/R K/R
___ ___ ___ ___ ___ ___
1 2 3 4 5 6
~O ~O ~K ~N ~N
~T ~K ~R ~F ~T
~K ~R ~F
~R

We can summarize all the deductions regarding the order of the books in the following diagram:

N/F N/F/T T/F O K/R K/R
___ ___ ___ ___ ___ ___
1 2 3 4 5 6

Now, the question is asking us if O is summarized, which of the following CANNOT be true?

Let's consider each of the answer choices:

(A) F is the first book discussed;

This is Option 2 outlined above. F could be first, N could be summarized and O summarized - this order could be true and no rules are violated.

(B) K is the 6th book discussed;

We can see from our setup that K is either #5 or #6 so this could definitely be true.

(C) F is summarized;

Let's think what happens if F is summarized. Notice that F is either #1, #2 or #3. Let's consider each of these options:

Scenario A: F is #3

F cannot be #3 because we know that O is #4 and is summarized and thus it would violate rule (1).

Scenario B: F is #1 or #2

If F is #1 or #2, F & N become consecutive because N must be #1 or #2. So if F is summarized, it MUST BE TRUE that N is not summarized or it again would violate rule (1).
s s
N/F N/F T O K/R K/R
___ ___ ___ ___ ___ ___
1 2 3 4 5 6

Rule (2) tells us that if N is not summarized - T is summarized, but this is impossible and would violate rule (1) because T in this scenario MUST be #3 and T & O become consecutive.

(C) is the correct answer choice.

(D) K is not summarized;

K could be #5 or #6 and not summarized. There are no rules that say otherwise.

(E) N is not summarized;

Per rule (2), R & T must be summarized so we need to consider if an order is possible where neither R nor T are adjacent to O:

N T F O K R
___ ___ ___ ___ ___ ___
1 2 3 4 5 6

This is a valid order of books meaning (E) could be true.

Does this make sense?
Let me know if you have any further questions.