Conundrum: Prospero’s Books
Jack Prospero buys individual hard-bound volumes of 31 different Shakespeare plays and an empty six-shelf bookcase to put them in.
He puts 3 plays each on the first and second shelves. He puts 5 plays each on the third and fourth shelves. He puts 7 plays on the fifth shelf, and 8 plays on the sixth shelf.
Within each shelf, the plays are in alphabetical order. The titles are exactly as they appear on this list. Ignoring any leading “The” or “A” articles, they are alphabetized by these exact titles.
And, as it turns out, the plays within each shelf are also in exactly the same order as they appear throughout that very same list!
One of the six shelves has only plays with the letter “F” somewhere in the title. A different shelf has no plays with any punctuation marks in the title. One shelf has more than half of its plays containing the word “King” in the title. Pairs of shelves with the same number of books in each are ordered alphabetically by first title.
Can you list the plays as they appear on each shelf?
UPDATE: Puzzle solved by ArtVark. See comments for answer.
September 3rd, 2012 at 4:02 pm
Shelf 1:
Famous History of the Life of King Henry the Eighth
Hamlet
King Lear
Shelf 2:
Measure for Measure
Merchant of Venice
Tragedy of King Richard the Second
Shelf 3:
As You Like it
Life and Death of King John
Life of King Henry the Fifth
Romeo and Juliet
Timon of Athens
Shelf 4:
Merry Wives of Windsor
Much Ado about Nothing
Taming of the Shrew
Twelfth-Night; or, What you Will
Winter’s Tale
Shelf 5:
All’s Well that Ends Well
First Part of King Henry the Fourth
First Part of King Henry the Sixth
Julius Caesar
Macbeth
Othello, the Moor of Venice
Pericles, Prince of Tyre
Shelf 6:
Comedy of Errors
Love’s Labours Lost
Midsummer Night’s Dream
Second Part of King Henry the Fourth
Second Part of King Henry the Sixth
Third Part of King Hery the Sixth
Tragedy of King Richard the Third
Troilus and Cressida
September 3rd, 2012 at 5:21 pm
Absolutely correct. Nice solving, ArtVark!
September 8th, 2012 at 5:06 pm
For this puzzle, a proof wasn’t required, as you can see that the answer fits all of the stated rules. But I needed to write a proof so that I knew the answer was unique, so I’ll share it with you now.
First. let’s define a “valid path” as a subset of the larger list of titles that is already in alphabetical order. All six shelves, then, may only contain books that form a valid path. We can prefix the word with a number to describe the length of a path. So a 5-path is a valid path of length 5.
To fill the sixth shelf, we must find a valid 8-path. This can be a challenge in and of itself. One way to approach it is to assign each play title a number to indicate the latest it may appear in any valid path.
The first title on the list is The Tempest. We’ll assign that a number 1, since it must be first on any valid path it is in. The second play on the list, The Two Gentlemen of Verona, has a number of 2. It can be as late as 2nd on a valid path, as long as The Tempest is also in the path. The Merry Wives of Windsor is a 1. Neither of the two plays that precede it on the list also precede it alphabetically.
We then continue down the larger list. To determine a play’s number, consider the set of titles that both precede that play on the list and alphabetically. We will already have determined all of their numbers; which is the highest? Add 1 to that, and you have the number for that play.
Once you’ve done this for the entire list, you find that only one play has a number of 8, and none have a number higher than 8. Therefore, the sixth shelf must end with Troilus and Cressida. And there is only one choice point in getting there. Using this method, it can be determined that the sixth shelf is as follows:
SIXTH SHELF
6.1. The Comedy of Errors
6.2. Love’s Labour’s Lost
6.3. Either The Merchant of Venice or A Midsummer Night’s Dream
6.4. The Second Part of King Henry the Fourth
6.5. The Second Part of King Henry the Sixth
6.6. The Third Part of King Henry the Sixth
6.7. The Tragedy of King Richard the Third
6.8. Troilus and Cressida
We now know 7 of the 8 books on the sixth shelf, and have the other book narrowed down to two options.
September 8th, 2012 at 5:16 pm
The next step is to find the books on the fifth shelf. To do this, we remove from the list the 7 books we know are on the sixth shelf, and we use the same method to find a 7-path from among the remaining titles.
Again, the options are limited. The fifth shelf must be some combination of the following books:
FIFTH SHELF
5.1. Either All’s Well that Ends Well or As You Like It
5.2. The First Part of King Henry the Fourth
5.3. First Part of King Henry the Sixth
5.4. Either Hamlet, Prince of Denmark or Julius Caesar
5.5. Either King Lear or Macbeth
5.6. Othello, the Moor of Venice
5.7. Pericles, Prince of Tyre
Also note that we can eliminate all combinations that include both 5.4. Hamlet, Prince of Denmark and 5.5. Macbeth, because Macbeth appears above Hamlet on the list.
Next, we eliminate the four books we know are on the fifth shelf and try to find two disjoint 5-paths from what remains. As it turns out, there are only two sets of 5-paths that are valid, and they are completely disjoint. We don’t yet know their first titles, but we know enough to put them in order.
THIRD SHELF
3.1. Either All’s Well that Ends Well or As You Like It
3.2. The Life and Death of King John
3.3. The Life of King Henry the Fifth
3.4. Romeo and Juliet
3.5. Timon of Athens
FOURTH SHELF
4.1. Either Measure for Measure or The Merry Wives of Windsor
4.2. Narrowed down to The Merchant of Venice, A Midsummer Night’s Dream, and Much Ado About Nothing
4.3. The Taming of the Shrew
4.4. Twelfth Night; or, What You Will
4.5. The Winter’s Tale
Also, we can eliminate all combinations that include both 4.1. The Merry Wives of Windsor and 4.2. The Merchant of Venice, because Merchant comes before Merry alphabetically.
We can now eliminate all of the titles that we have demonstrated to be on the third shelf and the fourth shelf from our shrinking list. We can also eliminate All’s Well that Ends Well and As You Like It, since we know that one of them is on the fifth shelf and the other one is on the third shelf.
Again, there are only two sets of 3-paths possible, and they are disjoint. And again, we have enough information to put them in order. The paths are as follows.
FIRST SHELF
1.1. Either Coriolanus or The Famous History of the Life of King Henry the Eighth
1.2. Either Hamlet, Prince of Denmark or Julius Caesar
1.3. Either King Lear or Macbeth
And remember, Hamlet and Macbeth cannot be on the same list. Since these combinations appear on two lists, we can conclude that one of the lists has Hamlet and King Lear and the other list has Julius Caesar and Macbeth.
SECOND SHELF
2.1. Either Measure for Measure or The Merry Wives of Windsor
2.2. Narrowed down to The Merchant of Venice, A Midsummer Night’s Dream, and Much Ado About Nothing
2.3. Either Titus Andronicus or The Tragedy of King Richard the Second.
And remember, The Merry Wives of Windsor and The Merchant of Venice cannot be on the same list.
September 8th, 2012 at 5:24 pm
We’re told that at least one of the shelves has more than half of the plays with “King” in the title. At our present level of progress, what’s the highest ratio possible for each shelf?
First: 2/3
Second: 1/3
Third: 2/5
Fourth: 0/5
Fifth: 3/7
Sixth: 4/8
The only shelf that could possibly have more than half is the first shelf, which means that it must be that shelf. Therefore, it must have two out of three plays with “King” in the title. This means that King Henry VIII is on the shelf and King Lear is on this shelf, which means that Hamlet must be as well.
FIRST SHELF
1.1. The Famous History of the Life of King Henry the Eighth
1.2. Hamlet, Prince of Denmark
1.3. King Lear.
With Hamlet and King Lear taken, 5.4.must be Julius Caesar and 5.5. must be Macbeth.
We’re told that at least one of the shelves has plays that all contain an “F” in the title. Which shelves do we know for sure contain at least one F-less play?
First: King Lear
Second: N/A
Third: Romeo and Juliet
Fourth: The Winter’s Tale
Fifth: Macbeth
Sixth: Troilus and Cressida
Therefore, the shelf with F-only titles must be the second shelf. 2.3. must be The Tragedy of King Richard the Second and 2.2. must be The Merchant of Venice. Since The Merchant of Venice can’t be on the same shelf as The Merry Wives of Windsor, then 2.1. must be Measure for Measure and 4.1. must be The Merry Wives of Windsor. 6.3. is therefore A Midsummer Night’s Dream, and so 4.2. must be Much Ado about Nothing.
We are told that at least one of the six shelves has no plays with any punctuation marks in the title. Which shelves do we know for sure contain at least one title with punctuation marks?
First: Hamlet, Prince of Denmark
Second: N/A
Third: N/A
Fourth: The Winter’s Tale
Fifth: Othello, the Moor of Venice
Sixth: Love’s Labour’s Lost
We can eliminate the second shelf, because we are told that it is a different shelf from the all-F shelf. So the non-punctuation shelf identified in the problem must be the third shelf. So 3.1. must be As You Like It, which means that 5.1. is All’s Well That Ends Well.
We are now left with only one possible solution, the one listed by ArtVark above.