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Treebeard's Stumper Answer
17 March 2000

Harry Potter's Puzzle

Near the end of J. K. Rowling's Harry Potter and the Sorcerer's Stone, Hogwarts' students Harry and Hermione are trapped in a room with both exits blocked by magic fire. There are seven bottles on a table. One is a potion to go forward, and another takes you back. The other bottles hold either wine or poison. There's also a note with clues, which I've reproduced below. "This isn't magic -- it's logic!" Hermione quickly finds the two potions they need, but I was disappointed. She can see the size of the bottles, but we can't. Was Hermione lucky, or was her logic really enough to solve the puzzle?

Harry and Hermione are trapped. There's a table with a row of seven different-sized bottles (not in order), and the following note:

Danger lies before you, while safety lies behind,
Two of us will help you, whichever you would find,

One among us seven will let you move ahead,
Another will transport the drinker back instead,

Two among our number hold only nettle wine,
Three of us are killers, waiting hidden in line.

Choose, unless you wish to stay here forevermore,
To help you in your choice, we give you these clues four:

First, however slyly the poison tries to hide
You will always find some on nettle wine's left side;

Second, different are those who stand at either end,
But if you would move onward, neither is your friend;

Third, as you see clearly, all are different size,
Neither dwarf nor giant holds death in their insides;

Fourth, the second left and the second on the right
Are twins once you taste them, though different at first sight.

Spoiler ahead! Just cross out every second letter in each word below to read Hermione's correct answer. But how did Hermione figure this, and which bottle (1 - 7) is the smallest?

Txhdew pqoytuihonnl itnf tdhsef srmpaklflsevsztq bvobtntulmer wyiyljlp tqaakfes thhgeumk fdoqrbwmakrsdg, arnfdg tjhwep rridgjhktsmbonshtz bbottrthljek wdisljlp tyatkkex tfhaebmp bmawcgky.

Hermione can solve the logic puzzle because she can see the size of the seven bottles, even though we readers can't. But she can only deduce her answer if the smallest bottle is either #3 or 4 (from the left), and the largest bottle is #2 or 6. Then she knows that the ordering must be:

{Poison, Wine, ?, ?, Poison, Wine, Back}

And the potion to go forward must be whichever of #3 or 4 is the smallest. The details are below. The real stumper is why author J. K. Rowling didn't include a picture so that we could figure it out too!

One of the eight arrangements of bottles that makes this puzzle solvable.
Bottle #3 is the dwarf, and #6 is the giant. Can you find the potions now?

The Gory Details of Harry Potter's Puzzle:

"Brilliant" said Hermione. "This isn't magic -- it's logic -- a puzzle. A lot of the greatest wizards haven't got an ounce of logic, they'd be stuck here forever... Everything we need is here on this paper. Seven bottles: three are poison; two are wine; one will get us safely through the black flame; and one will get us back through the purple... Give me a minute..."

Here's what we know:

In addition, there are these clues and corollaries:

Clue 1: Every W has a P to it's left.
  • #1 is not W.
  • Two of the three Ps have a W on their right. The other one doesn't.

  • Clue 2: The end bottles (#1 and #7) are different, and neither is F.
  • #1 is not W or F. It must be P or B.

  • Clue 3: The largest and smallest bottles (+/-) are not P.

    Clue 4: #2 and #6 are the same.
  • #2 and #6 must be W or P since there is only one of each potion.

  • Altogether there are 7!/(3! 2!) = 420 arrangements of the bottles. But the clues narrow it down. The question is whether the clues narrow it down to just one correct solution. Since we readers can't see the bottles, we have to begin without clue 3. The most useful clue to start with is the corollary of clue 4, that #2 and #6 must be W or P. We'll consider them both in turn.

    Case 1:    #2 and #6 are both W.

    Then #1 and #5 are both P (clue 1). Bottle #7 can't be W since we already have them both, and it can't be P since it must be different from #1 and it can't be F (clue 2). Therefore #7 must be B. So the arrangement must look like this, where the unknowns are either F or P:

        P     W     ?     ?     P     W     B    

    This gives two possible arrangements that match the clues:

    (a1)   P     W     F     P     P     W     B    
    (a2)   P     W     P     F     P     W     B    

    It's tempting to assume that #2 and #6 must be wine since clue 4 mentions tasting them. Graybear assumed this, as did Whitney and Ilana and Samo at Dunn Middle School. That simplifies things, but it's not right. It might also mean that both are deadly poison with a delayed effect. We have to consider case 2:

    Case 2:    #2 and #6 are both P.

    So there's another P hiding somewhere else. By clue 2, at least one of #2 and #6 must have a W to the right, so there are 3 possibilities to consider ("W" is not-W):

    2a:         P     W                 P     W    
    2b:         P     W                 P     W    
    2c:         P     W                 P     W    

    Case 2a:    #7 is not W.

    2a:         P     W                 P     W    

    So #5 must be W so that #4 can be the third P on its left. But then #1 or #7 must be F, which violates clue 4. Case 2a is not possible.

    Case 2b:    #3 is not W.

    2b:         P     W                 P     W    

    There's two possibilities for the other W, either #4 or #5. Each choice brings along the third P to it's left, and leaves B and F to place. Since #1 can't be F (clue 2), it must be B. This leaves two choices:

    (a3)   B     P     F     P     W     P     W    
    (a4)   B     P     P     W     F     P     W    

    Case 2c:    #3 and #7 are both W.

    2c:         P     W                 P     W    

    Bottle #1 can't be F (clue 2), so it must be either B or P. This leaves 4 choices:

    (a5)   B     P     W     F     P     P     W    
    (a6)   B     P     W     P     F     P     W    
    (a7)   P     P     W     F     B     P     W    
    (a8)   P     P     W     B     F     P     W    

    That's it! There are just these eight arrangements that satisfy rules 1,2, and 4:

    (a1)   P     W     F     P     P     W     B    
    (a2)   P     W     P     F     P     W     B    
    (a3)   B     P     F     P     W     P     W    
    (a4)   B     P     P     W     F     P     W    
    (a5)   B     P     W     F     P     P     W    
    (a6)   B     P     W     P     F     P     W    
    (a7)   P     P     W     F     B     P     W    
    (a8)   P     P     W     B     F     P     W    

    That's the best we can do without seeing the bottles. One chance in eight is better than one in 420, but it's not good enough.

    With clue 3, we know that the largest and smallest bottles are not P. So for Hermione, they must have been enough to eliminate all doubts. The only way this could be is if one of the extreme bottles is #2 or #6, but not both, since that would be redundant by clue 4. Either way, we can eliminate (a3) to (a8), all of Case 2 with P in #2 and #6. With only the first two possibilities of Case 1 left, we know that the Back potion must be #7 and the Forward potion must be the other extreme bottle. With any other arrangement, Hermione (the perfect logician) couldn't solve the puzzle.

    Since Hermione was sure of the solution, the arrangement must have been one of these eight variations of soultions (a1) and (a2). The biggest (+) and smallest (-) bottles are placed as shown. In all solutions, one of the smallest or largest bottles is either #3 or #4, and the opposite-sized bottle is #2 or #6. Then Hermione knows what the ordering must be. Any of these arrangements could be the illustration for the stumper in the book. Any other would fail.

    (s1)   P     W+     F-     P     P     W     B    
    (s2)   P     W-     F+     P     P     W     B    
    (s3)   P     W+     P     F-     P     W     B    
    (s4)   P     W-     P     F+     P     W     B    
    (s5)   P     W     F+     P     P     W-     B    
    (s6)   P     W     F-     P     P     W+     B    
    (s7)   P     W     P     F+     P     W-     B    
    (s8)   P     W     P     F-     P     W+     B    

    This was tedious and complicated, but it would be easier for Hermione. Look at the picture again:

    The sixth bottle is the giant, so the second and sixth are the same (clue 4). They don't hold poison (clue 3), so they must both contain nettle wine. Therefore the first and fifth must hold poison. The last bottle on the right can't hold poison or the forward potion (clue 2), and the wine is all accounted for, so it must hold the back potion. The third bottle is the dwarf, so it can't be poison (clue 3), and so it must be the forwards potion. It's like this:

      P     W     F-     P     P     W+     B  

    It's a piece of cake!


    It's unusual to work backwards from the fact that there is a solution to figure what the question must have been, but there are other puzzles that work the same way. (For example?)

    Graybear and DMS students Whitney, Ilana, and Samo all got the right answer, but they all assumed that #2 and #6 are wine. In fact, you don't need to assume that, you can prove it.

    Graybear suggests adding a fifth clue:

    Fifth, different are those who are next to each other,
    For the sake of the blind, this reduces the bother.

    Cute, but it misses s1, s2, s5, and s6.

    Why did author J. K. Rowling create this fine puzzle and then have Hermione announce the solution with no opportunity for us to solve it? Well, maybe she gave us a better stumper in its place!

    Maybe that was the author's intent, but I worry that some editor cut this puzzle on the assumption that the intended audience of kids couldn't handle it. DMS students proved otherwise! That same editor probably changed the name of the American edition from the original British Harry Potter and the Philosopher's Stone. I can just imagine the conversation: "American kids will never buy a book with the word philosopher in the title." It's not the kids who are dumb!

    Another mystery is why you-know-who didn't drink all the forward potion, or take it with him/her/it, or scramble the order of the bottles into an unsolvable position. Sometimes the logic of a story takes precedence over the logic of a wizard!

    Here are some Web links for further inquiry:

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