With the exception of Q the books P Q R and S

Question

With the exception of Q, the books-P, Q, R, and S-on Ethan's desk are such that the page numbered 1 is on the right of the book. Ethan kept an important note in one of the books. Later, he forgot which book it was-but he accurately recalled having placed the note between pages 536 and 537 of a book with an entirely orange cover. Ethan knew the page numbering scheme and the number of pages in all four books. He also knew that in each of the books, the pages are numbered consistently and consecutively throughout the book. How many books did Ethan need to check to find the note?

(1) Each of P and S has fewer than 500 pages.
(2) P and S have entirely blue covers.

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Official Answer

D

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XLmafia21 · Accepted Answer

Q has 1 on the left page.. so pages will appear from the start as 1-2, 3-4, 5-6, 7-8.. point to note here is that any item kept in between two pages will include an odd numbered page and even numbered page in sequence.
P,R,S has 1 on the right side.. so pages will appear from the start as 1, 2-3, 4-5, 6-7,... here any item kept in b/w two pages will include an even numbered page and odd numbered page in sequence.
What we know from the stem is that the note is kept in between 536-537 so it has to be one among P, R and S.

1. If P and S both have fewer than 500 pages, then only candidate to be checked is R. so, one book.
2. If P and S both have blue cover and note is in orange cover book, then P and S are again eliminated and we are left to check only book R. So, one book.

option D.

iBN · Answer

The way I solved is

4 Books, but Q has page 1 on left while others have it on right

In books PRS the mentioned page numbers should be on one spread. 536 ON left and 537 on right side.

Option 1 - P AND S can be skipped. so 2 books must be checked which is R and Q. (i know in real life in Book Q the spread will not be on the same page ...bit of a stretch to consider Q imo).

Option 2 - Again P AND S can be skipped. as they have blue and the person is certain that its an orange cover. hence again 2 books must be considered.

so i chose option D.

I was confused between option B and D. (After marking the question i asked gemini ai - it has marked E ) so waiting for an expert to answer this

ivycation2024 · Answer

This is a brilliant question, I was missing out on what  "With the exception of Q, the books-P, Q, R, and S-on Ethan's desk are such that the page numbered 1 is on the right of the book."  would mean for the answer!

Utsharma · Answer

This seems like a question for a private investigator entrance exam

roshaun25 · Answer

Why would each statement be sufficient here? Each statement only addresses one of the two limiting factors (color and number of pages), so in my opinion neither statement on its own if sufficient. Let's say that statement 1 is true - at first look you might say that Q and R need to be checked. But what if book R has a green cover? Then only Q needs to be checked. Each statement only addresses a single factor, so I'm confused how they are each sufficient.

Bunuel · Answer

With the exception of Q, the books-P, Q, R, and S-on Ethan's desk are such that the page numbered 1 is on the right of the book. Ethan kept an important note in one of the books. Later, he forgot which book it was-but he accurately recalled having placed the note between pages 536 and 537 of a book with an entirely orange cover. Ethan knew the page numbering scheme and the number of pages in all four books. He also knew that in each of the books, the pages are numbered consistently and consecutively throughout the book. How many books did Ethan need to check to find the note?

We need the number of books that could possibly match Ethan’s accurate memory: an entirely orange cover, and the note placed between pages 536 and 537.

Key fact from the stem: In books where page 1 is on the right (P, R and S), odd pages are on the right and even pages are on the left. So pages 536 (even) and 537 (odd) are facing pages, so placing a note between them makes sense. In Q, page 1 is on the left, so odd pages are on the left and even pages are on the right, which makes pages 536 and 537 not facing pages. So  Q cannot be the book .

(1) Each of P and S has fewer than 500 pages.

Then P and S cannot have pages 536 and 537. Q cannot be the book as explained above. So only R can possibly contain the note, so Ethan only needs to check 1 book. Sufficient.

(2) P and S have entirely blue covers.

The note is in an entirely orange cover book, so P and S are eliminated by color. The only remaining possibilities by color are Q and R. But Q cannot be the book (pages 536 and 537 are not facing in Q). So only R remains, and Ethan only needs to check 1 book. Sufficient.

Answer: D.

Bunuel · Answer

Because the stem already eliminates Q on its own.

If page 1 is on the right, then odd pages are on the right and even pages are on the left, so pages 536 and 537 face each other and you can put a note “between” them.

But Q is the exception: page 1 is on the left, so odd pages are on the left and even pages are on the right. That means pages 536 and 537 are not facing pages in Q, so the note cannot be “between pages 536 and 537” in Q. So Q is impossible before you even look at (1) or (2).

Now look at each statement:

(1) P and S have fewer than 500 pages.

Then P and S cannot even have pages 536 and 537. Q is already impossible from the stem. So the only remaining book is R, so Ethan only needs to check 1 book.

(2) P and S are entirely blue.

Ethan remembers the cover was entirely orange, so P and S are eliminated by color. The only remaining candidates by color would be Q and R, but Q is already impossible from the stem. So only R remains, so again he checks 1 book.

Your “what if R is green” idea is not allowed because Ethan’s recollection is stated to be accurate. If the only possible book left were not orange, that would contradict the facts of the problem.

egmat · Answer

You're right that we need to track both color and page count - but there's a  third constraint  hidden in the question stem that you may have overlooked.

Book Q is automatically eliminated - not because of color or page count, but because of its page numbering

Here's why Q can never contain the note:

The question tells us:  "With the exception of Q, the books... are such that the page numbered 1 is on the right."

This means:
 
  Books P, R, S (page 1 on right):  Odd pages appear on the right, even pages on the left. So page 536 (even) is on the left, page 537 (odd) is on the right. These pages  face each other  - you can slip a note between them. 
  Book Q (page 1 on left):  Odd pages appear on the left, even pages on the right. So page 536 (even) is on the right, page 537 (odd) is on the left. These pages are on  opposite sides of the same leaf  (back-to-back) - you  cannot  place a note "between" them!

So before we even look at the statements, Q is already eliminated

With the exception of Q the books P Q R and S

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