Problem 902

Near the end of the book "Harry Potter and the Sorcerer's Stone",
by J. K. Rowling, Hogwarts students Harry Potter and his friend
Hermione Granger find themselves in a room with two doors.
One door leads forward to a place they are trying to get to; the
other leads back to where they have just come from, but both
doorways are blocked by flames. In the room there is a
table with a row of seven bottles on it, 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."

Hermione studies the note and the row of bottles and comes to a
definite, and correct, conclusion about which bottle will carry
them forward  and which will allow them to go back. What is the
position of the bottle that allows them to move back?

Suggested by Dan Velleman, Amherst College

-----------------------------------------------------------------------------------------

Problem 902 Solution:
We received solutions from David Castro (Retek) and Bobby Schwalbe
(an 8th grader at Wayzata, MN, Central Middle School).

Answer: Position 7.

Use the following notation:
P = poison
W = wine
F = forward
B = back
E = giant or dwarf bottle

The idea of this solution is to consider the possible location of the Es

in order for Hermione to come to a definite, correct conclusion.

CASE 1: The E's are located in positions {2, 3} or {2, 4} or {3, 6} or
{4, 6}.

In this case, she knows that the ordering must be   P W ? ? P W B.
If position 2 is E, then she knows that F is in position 2.
If position 3 is E, then she knows that F is in position 3.

We rule out the rest of the possibilities.

CASE 2: The E's are in positions {2, 6}.

Then we must have the configuration P W ? ? P W B and there is no way
to determine where F is placed.

CASE 3: The Es are in positions {2, 5} or {6, 5}.

These contradict the rule that P is left of W.

CASE 4: There is an E in position 1.

Then we must have B P ? ? ? P W, and there is only one remaining E, so
there is no way to determine where the F is.

CASE 5: There is an E in position 7.
CASE 6: The positions of the Es are in  {3, 4, 5}.

Rule these out by trying all possibilities.