The Riddle of the Potions

The Riddle of the Potions seldom gets the respect it merits. It is the last challenge faced by Harry and Hermione before entering the chamber where the Philosopher’s Stone is kept safely hidden in the Mirror of Erised. Usually when reading the book, we speed past the riddle, admire Hermione for solving it, enjoy Harry and Hermione’s parting scene, and then race on the next chapter for the book’s climax. The riddle is left in the dust.

I have often wondered if the riddle was for real. Can one solve the puzzle given the clues available? To put it another way, did Rowling really put forth the effort to devise a real puzzle, or did she just put together some rhyming verses that sounded good, without any real thought of working them out? After all, there are some 420 possible solutions to a puzzle involving 3 poisons, 2 wines, 1 forward potion, and 1 backward potion.

I am happy to say that it can be logically proven that she did put forth the time and effort to create a real puzzle. The riddle leads inexorably to two and only two possible solutions. If we knew the location of the smallest bottle, we would be able to narrow it down to one.

First some definitions:
CLUES: The actual wording of the riddle. These will be numbered as CLUE #.
RULES: Facts derived directly from the clues. These will be numbered as R#.
COROLLARIES: Facts derived from the rules. These will be numbered as C#.

The bottles will be number one through seven, beginning on the left. They look like this:
Left 1 2 3 4 5 6 7 RIGHT

So let us begin.

CLUE 1: “One among us seven will let you move ahead.”
This tells us that there is one and only one potion that will allow us to go forward. We shall call it F.

CLUE 2: “Another will transport the drinker back instead.”
This tells us that there is one and only one potion that will allow us to go backwards. We shall call it B.

CLUE 3: “Two among our number hold only nettle wine.”
Two bottles contain nettle wine. These we shall call W.

CLUE 4: “Three of us are killers, waiting hidden in line.”
Three bottles contain poison. These we shall call P.

CLUE 5: “First, however slyly the poison tries to hide — You will always find some on nettle wine’s left side;”
R1: Whenever there is a wine, you will always find poison to its left.
C1: Bottle #1 cannot be a wine.
C2: The three poisons cannot be together. That would leave at least one wine without a poison to its left.
C3: Two poisons may be next to each other; but in that case, there must be a wine immediately to their right.
C4: If Bottle #1 is not poison, Bottle #2 cannot be wine.

CLUE 6: “Second, different are those who stand at either end, But if you would move onward, neither is your friend;”
R2: Bottles #1 and #7 never contain the same potion.
R3: Bottle #1 is not F.
R4: Bottle #7 is not F.

CLUE 7: “Third, as you see clearly, all are different size, Neither dwarf nor giant holds death in their insides;”
R5: The biggest bottle is not poison.
R6: The smallest bottle is not poison.

CLUE 8: “Fourth, the second left and the second on the right — Are twins once you taste them, though different at first sight.”
R7: Bottle #2 and Bottle #6 hold identical potions.
C5: Bottle #2 is not F, since F is unique.
C6: Bottle #6 is not F, since F is unique.
C7: Bottle #2 is not B, since B is unique.
C8: Bottle #6 is not B, since B is unique.

Putting It Together:
Let’s first look at Bottle #1. From C1 and R3 we know that it must be either P or B. Let’s first assume that it is B. From C4, C5, and C7 Bottle #2 must be P. Using R7, Bottle #6 is P as well. Using C3, Bottle #7 cannot be a poison. Since it cannot be B (already at Bottle #1), P, or F (from R4), it must be a W. So all possible solutions with Bottle #1 being B take the form of : B P 3 4 5 P W.

If 3 is W, then R1 is satisfied by bottles #2, #3, #6, and #7. Bottles #4 and #5 can be F P or P F. Thus two possible solutions are:
S1: B P W P F P W
S2: B P W F P P W

Going back to B P 3 4 5 P W, if 4 is W, then 3 must be P (R1) leaving 5 to be F. If 5 is W, then 4 must be P (R1) leaving 3 to be F. Thus two more possible solutions are:
S3: B P P W F P W
S4: B P F P W P W

That exhausts all the possible solutions if Bottle #1 were B. Now let’s look at the solutions if Bottle #1 were P (C1,R3). We know that Bottle #2 must be P or W (C5, C7). If it is P, then so must Bottle #6 (R7). This forces bottles #3 and #7 to be W (R1). The two possible solutions here are:
S5: P P W F B P W
S6: P P W B F P W

Now if Bottle #2 was a W, then #6 would be also (R7), forcing Bottle #5 to be P. Bottle #7 cannot be a W (there are only two and they are at #2 and #6.), F (R4), or P (R2). It must be B. This leaves the final possible solutions with Bottle #1 being P as:
S7: P W P F P W B
S8: P W F P P W B

In summary, the eight possible solutions are:
S1: B P W P F P W
S2: B P W F P P W
S3: B P P W F P W
S4: B P F P W P W
S5: P P W F B P W
S6: P P W B F P W
S7: P W P F P W B
S8: P W F P P W B

Not bad for starting out with 420 possible solutions!

This is the best we can do with the clues given. R5 and R6, which tell us that the largest and the smallest bottles are not poison, cannot be used directly since Rowling does not tell us the relative sizes of the bottles. However, if we work backwards from the solution provided by Hermione, we can narrow the field to the most likely two solutions.

Consider R5 and R6. Rowling does not tell us the different sizes of the bottles, other than inform us later that F is the smallest bottle. We also learn from Hermione that Bottle #7 is B. What can we infer from that?

Note that there are two and only two solutions that have Bottle #7 being B. These are also the only two solutions that do not have P for Bottles #2 and #6. Therefore, if either Bottle #2 or #6 were the largest bottle, we are left with only two possible solutions:
S7: P W P F P W B
S8: P W F P P W B

Notice the only difference between the two solutions is the swapping of Bottles #3 and #4. One is F, the other is a P. If either one was the smallest, we know it can’t be poison (R6). That forces the smallest bottle to be F.

So our solution looks like this:
1: Poison
2: Wine. The largest bottle if #6 is not.
3: Poison or Forward Potion if the smallest bottle.
4: Forward Potion if the smallest bottle, or Poison if not.
5: Poison
6: Wine. The largest bottle if #2 is not.
7: Back Potion

Conclusions:
We have seen that the Riddle of the Potions can come down to one of two solutions, depending upon the placement of the smallest bottle. This is strong evidence that Rowling did indeed work out all the parts of the riddle. She likely left the crucial size placement out in order to tighten up the story. She was writing an adventure story, not a puzzle book.