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The Rowling Magazine Issue #2 · December 2016

The Potions Puzzle

1825 words

It is the last challenge they faced, but people underestimate it (the film deleted this scene, so that could also be one of the reasons). Almost no magic, only a logical riddle crafted by Severus Snape, who is presented not only as a Potion Master but also as a rational man. There are a few questions this riddle presented us with, and we will try to discuss them – and along the way learn some mathematics!

The first one is how many arrangements there were in the Riddle of the Potions (Pottermore calls it “The Potions Puzzle”, by the way). Let’s see the original puzzle they found:

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.

There are seven bottles on the table. That is 5040 possible orders. Why? Well, that is because the notion of permutation, part of the combinatorics branch of mathematics. Let’s see the logic behind this.

If we have seven bottles and we want to sort them, in the first place we can select one of seven. That’s seven possibilities. After we select the first one, we can select one of six. That is six possibilities – but that is after we select the first one, so it would be six for each one of the initial seven. After the second one, we are left with five possibilities – but we already have selected the first two.

So, for the first position, we have 7 possibilities. For each one of these, we have 6. And for each one of these 6, we have 5. We can calculate all the possibilities with 7 x 6 x 5 x 4 x 3 x 2 x 1, and that is factorial of 7, or (7!). And the result is 5040.

However, we can reduce that number. We know there are three bottles of poison, two of wine, one to go forward, and one to go back. Let’s assume that there is no difference between the bottles of the same kind, so the possibilities are reduced since it is the same if we internally change them. (For example, if the first two bottles are of wine, it does not matter which one is the first and which one is the second).

So, we must divide our original number by (3! x 2!), because we have 3 bottles of poison and 2 bottles of wine – and we do not want to count their permutations twice. So (3! x 2!) = (6 x 2) = 12, and 5040 / 12 = 420, and that is in how many ways we can arrange the seven potions in the table.

Now we know Harry and Hermione could have faced 420 different scenarios, Let’s focus on the one they did. And mostly, is the puzzle solvable? And if it is, how?

We know there are 7 potions, two of them wine, three of them poison, one going back and one going forward. Let’s call them W (wine), P (poison), B (going back) and F (going forward). We must arrange them. One possibility could be W W P P P B F, but we know that is not going to work.

Let’s go back to the puzzle text. Besides the introduction, it provides four clues:

Clue 1: However slyly the poison tries to hide, you will always find some on nettle wine’s left side.
Every Wine bottle has a Poison to its left.

Clue 2: Different are those who stand at either end, but if you would move onward, neither is your friend.
The first is different to the last one. And none of them is the Potion Forward.

Clue 3: As you see clearly, all are different size. Neither dwarf nor giant holds death in their insides.
We are unable to take anything from this one – since we cannot see the potions!

Clue 4: The second left and the second on the right are twins once you taste them, though different at first sight.
The second and the sixth one are of the same kind.

With these four clues we should be able to solve the puzzle – at least Hermione was! We must note she had a little advantage: she was seeing the bottles, which is important for Clue 3: without knowing in which positions these potions are, we cannot take any fact from it.

From Clue 1 we know we have two pairs of the form PW, so we can consider we have now 5 entities to order: PW, PW, P, B and F. There are many ways this could be arranged, so we have to continue.

Clue 2 gives a lot of information. First of all, F could not be on either end of the table. Second, W could not be at the start, since it must have P on its left side. With this, we know the first potion in the table must be P or B.

If it is P, then the one at the right end is W or B. If the first one is B, the row ends with W or P. So we have these four possibilities so far:

P1: P X X X X P W
P2: P X X X X X B
P3: B X X X X P W
P4: B X X X X X P

Note that for P1 and P3, we also know the sixth potion. As we previously said, Clue 3 does not give us any information. But Clue 4 is very useful: it says that the second and sixth potion must be the same type. As Back and Foward are unique, they cannot be in those positions, so they must be poison or wine.

Let’s take Possibility 1: as the sixth one is P, the second one must be P too. And the only left place to its right must be W, so Possibility 1 becomes P P W X X P W.

With Possibility 2 we do not know what is in the sixth position, but using logic we can know. If position 6 contains Poison, then position 2 should have Poison too. And that would leave us with two Wines, and we would not be able to put them with a Poison to their left (one Poison for each one), so this cannot happen. Then, wine must be in position two and position six. And having wine at position six, Poison is placed at position five. Then, Possibility 2 becomes P W X X P W B.

In the case of Possibility 3 we only can add P to the second position, leaving us with B P X X X P W, but nothing else.

Possibility 4 would give us a little rest. We cannot put W in the second and sixth position, since the one in position 2 would not have P to its left. So we have to put P at both positions, but that would lead us to have P P on the last two positions. Then we could not match the two W to two P to have to their lefts. So Possiblity 4 cannot work and we must get rid of it.

Without seeing the table with the potions, this is how far we can get. We have three (incomplete) possibilities:

P P W X X P W, P W x x P W B and B P x x x P W. Since the first and second one can be divided into two possibilities each, and the third one into four possibilities, we end up with eight possibilities:

1. P P W B F P W
2. P P W F B P W
3. P W F P P W B
4. P W P F P W B
5. B P W P F P W
6. B P P W F P W
7. B P W F P P W
8. B P F P W P W

Without seeing it, we cannot use Clue 3 and that is how far we can get. However, we can continue the exercise to try to figure out which was the scenario Harry and Hermione faced (using a clue giving by the narration of the book), and how Hermione was able to resolve it.

We know Hermione identified successfully the potions, and she drank from the Back potion from the end of the line. That narrows the eight possibilities to only two, 3 and 4:

1. P W F P P W B
2. P W P F P W B

But can we know if Forward potion is in position 3 or 4? Well, the narration tell us Harry drank from it, and it was the smallest bottle in the table.

We know Hermione resolved the puzzle, so there must be something in Clue 3 that allowed her to choose one from the eight possibilities we considered above. That clue says the largest and the tiniest bottle is Wine, Forward or Back. If the largest potion had been the Back at the end of the row, it would have been impossible for Hermione to identify it. But if the largest had been in position 2 or 6 (both Wine), Hermione could have filtered the eight posibilities down to the two (because all the others have Poison in 2 and 6).

In that scenario, if the smallest potion was in position 3 or 4 (which apparently was), that could be only the Forward potion, and Hermione would have solved the puzzle, and also knew she had arrived at the right answer.

And that is how you can be in Hermione’s shoes and solve the puzzle that Snape put for them – or for anyone who wanted to get to the Philosopher’s Stone. It is hard to imagine that Quirrell/Voldemort were able to solve it too, maybe Voldemort knew a way to go through fire without using the potion.

But in case you are not as powerful as Voldemort, you may use all the logical power to decipher the riddle and keep on your path to the Mirror of Erised to get the Philosopher’s Stone.

You are reading an article from The Rowling Library Magazine Issue 2 (December 2016).
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