Answer to the October 7, 2002 Problem of the Week

The Harry Potter Problem   by Roger Howe   Mathematics Teacher February 2002

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In the middle of the climactic episode of J. K. Rowling's children's novel Harry Potter and the Sorcerer's Stone, Harry Potter and Hermione Granger find themselves in a room whose two exits are blocked by raging flames.

On a table in the room stand seven bottles, along with a piece of paper inscribed with the following verses:

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

(Rowling 1998, p. 285)

Hermione, who is an exceptionally avid student, realizes that she can use reasoning to determine the bottles that they need to escape from the room. As she says, "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 in here forever." Hermione thinks hard, and in a few minutes she is able to identify the bottles that allow escape. She goes back to sound the alarm, and Harry goes forward to confront the evil-doer.

Can you figure out the contents of each of the seven bottles below, just as Hermione did?

 

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Solution to the Problem:

Bottles 1, 4, and 5 are poison.
Bottles 2 and 6 are Nettle Wine.
Bottle 3 lets you move forward, and
bottle 7 lets you move back.

In the novel, you are not presented with a picture of the bottles and Hermione's reasoning is not explained. For a complete analysis of this problem, see Roger Howe's article in the February 2002 issue of the Mathematics Teacher.

Faced with the lineup of bottles shown above, Hermione would have known from the third clue that neither the largest (giant), at position 2, nor the smallest (dwarf), at position 3, could be poison. From the fourth clue, she would then know that the bottle in position 6, that is, second from right, also is not poison. The fourth clue says that bottles 2 and 6 are the same, so they must be nettle wine. Thus, bottles 1 and 5 must be poison by the first clue. Two positions, positions 4 and 7, then remain as possible locations for the third bottle of poison. The second clue indicates that bottles 1 and 7 are different, so 7 is not poison and 4 must be. Bottles 3 and 7 are then the elixirs of escape. Since the second clue says that the end bottle, bottle 7, will not help move onward, it must be the potion for returning. Therefore, bottle 3 is the one to drink to continue the quest.

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