Ravenclaw Tests of Logic


Brilliant, said Hermione. This isn't magic - it's logic.


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Snape's Puzzle

One of the most famous logic puzzles in recent wizarding history was the one devised by Professor Severus Snape to guard the Philosopher's Stone. This puzzle, solved by a first year Gryffindor student named Hermione Granger, was a challenge to find one of two potions hidden in a row of seven unmarked bottles. Professor Snape provided the following clues in the form of a poem:


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 hold 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.


The room had been set up with a charm such that magical flames blocked both the entrance and the exit. One of the seven potions would give the drinker the ability to safely move forward through the flames, while another would allow the drinker to go back safely. Of the remaining five, two were simply nettle wine and three were in fact deadly poison.

The problem facing those of us who would like to recreate Professor Snape's puzzle is that the puzzle relies in part on what the bottles look like - that is, the third clue talks about the smallest and largest bottle. We thus have only three clues to work with, while someone actually in the room would have had four. But let us try to work through the solution as best we can:

First Clue

Every bottle of nettle wine will have poison to its left. Now, this doesn't necessarily mean that every bottle of poison will have nettle wine to its right. Since there are three bottles of poison, and only two of nettle wine, that means there will be two "pairings" of nettle wine-plus-poison, with the wine on the right and the poison on the left. There will be one poison left over, as well as the forward potion and backward potion:

poison wine poison wine poison forward backward


There are many different ways that these could be arranged, so let's go on to the second clue:

Second Clue

There are two helpful bits of information here: first, that the potions on either end of the row are different from one another, and secondly that the "move forward" potion is not on either end.

Because the nettle wine always has to have poison to its left, the far left potion cannot be nettle wine. It could be poison or the move backward potion, though. If the far left one is poison, the far right one has to be either nettle wine or the move backward potion. If the far left one is the move backward potion, the far right one has to be either nettle wine or poison.

So that leaves us with four possible combinations for the ends (the potions are numbered 1 through 7 left to right):

1 2 3 4 5 6 7
Possibility A poison ? ? ? ? poison wine
Possibility B poison ? ? ? ? ? back
Possibility C back ? ? ? ? poison wine
Possibility D back ? ? ? ? ? poison


Note that for possibilities A and C, we would know what potion 6 is, because the wine on the right hand side has to have poison to the left of it.

Third Clue

This clue involves being able to see the potions - we do not know from the published accounts which bottles were the largest and the smallest. So we'll have to skip this clue for the moment.

Fourth Clue

This clue means that the potions in positions 2 and 6 have to be the same kind. Right off the bat, then, we know that neither one can be the move forward or move back potion, because neither one of those has a pair to be matched with. So 2 and 6 have to be both poison or both wine.

So now we can look at the four possibilities above and begin to figure out what the middle potions are. If we go with Possibility A, then the potion in position 2 has to be poison, to match the poison we know is in position 6. And there's the second nettle wine, too - we know that has to go to the right of one of the poisons, so it has to go in position 3 because that's the only spot left to the right of a poison. Positions 4 and 5, then, have to be the move forward and move backward potions, but we do not know which is which:

1 2 3 4 5 6 7
Possibility A1 poison poison wine forward back poison wine
Possibility A2 poison poison wine back forward poison wine


Now, if we go with Possibility B, we don't automatically know what's in position 6. But consider this - if position 6 were poison, then position 2 would have to contain poison as well. And that can't work, because there are still the two nettle wines, and there wouldn't be slots to the right of the poison to put the nettle wines in. So for Possibility B, both 2 and 6 have to contain nettle wine. Putting nettle wine in position 6 means poison has to be in position 5. The two left over are the move forward potion and the third poison, which have to be in positions 3 and 4. But just like Possibility A we can't know which is in which position:

1 2 3 4 5 6 7
Possibility B1 poison wine poison forward poison wine back
Possibility B2 poison wine forward poison poison wine back


If we go with Possibility C, things really turn into a mess. Just like Possibility A, the fact that poison has to be in position 6 means poison has to be in position 2 as well. But that's as far as this one takes us - there are three potions left over - the forward potion, the second nettle wine, and the third poison. Any one of them might be in any of the three unfilled positions (3, 4, and 5). A couple of the combinations won't work because the nettle wine wouldn't have poison to its left. But there are still four combinations that work:

1 2 3 4 5 6 7
Possibility C1 back poison wine poison forward poison wine
Possibility C2 back poison poison wine forward poison wine
Possibility C3 back poison wine forward poison poison wine
Possibility C4 back poison forward poison wine poison wine


Possibility D makes things a little better, though. For Possibility D, the potion in position 2 cannot be nettle wine because then the move backwards potion couldn't be in position 1. So position 2 has to have poison - and since position 2 and 6 have to be the same, position 6 would have to be poison as well. But if 2 and 6 are both poison, there would end up being a nettle wine that didn't have poison to the left of it. So Possibility D cannot work under any circumstances, and we can get rid of it.


So in the end, we're left with eight possible arrangements of the seven potions. In all of the arrangements, the move forward potion is in position 3, 4, or 5. So even without knowing the sizes of the bottles and using clue three, logic has allowed us to narrow down the possibilities a great deal.

Epilogue

Even though the clues can only take us so far, the published account of what happened as first year Gryffindor Hermione Granger solved the puzzle lets us narrow down the possibilities much further. Hermione, it is written, drank from the move backwards potion - and it says she took "a rounded bottle at the right end of the line."

Only the B Possibilities (both B1 and B2) have the move backwards potion at the far right end. So that eliminates all of the A and C possibilities:

1 2 3 4 5 6 7
Solution B1 poison wine poison forward poison wine back
Solution B2 poison wine forward poison poison wine back


But even down to these two possibilities, we do not know whether the move forward potion was in position 3 or 4. Without knowing more about the shapes of the bottles, logic can take us no further towards the solution to Professor Snape's puzzle.

However, there is one more piece of information from the published accounts. First year Gryffindor Harry Potter drank from the move forward potion, which was described as the smallest bottle. Now neither the smallest nor the largest were poison, so that means the largest bottle had to have been the move backwards potion or the nettle wine.

Now, although we do not know which was the smallest and which was the largest, we do know that using the other three clues only narrows the puzzle down to eight possibilities (A1, A2, B1, B2, C1, C2, C3, and C4). The only reason that we know that the solution has to be one of the B solutions is from what Hermione said after she'd already solved it. This means that there must have been something in that last clue that allowed Hermione to solve the puzzle.

If the largest bottle had been the move backwards potion, in position 7, Hermione would not have been able to know whether the largest bottle was the move backwards potion or nettle wine - it would not have narrowed down the possibilities enough. But if the largest bottle had been in position 2 or 6, Hermione would have been able to figure out that only Possibility B could be correct. And with that information, if the tiniest bottle were in position 3 or 4 (which it apparently was), it could only have been the move forward potion - which is why Hermione knew she'd arrived at the right answer. So although this added bit of information doesn't help us come to the full solution, it does tell us that the largest bottle was nettle wine and in position 2 or 6. And it gives us further insight into how Hermione solved the puzzle.


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