Philosopher's Stone Solution #2

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This is the shorthand I'll be using to discuss the problem:

A	The Ahead potion
B	The Back potion
P	A Poison potion
W	A bottle of Wine

Each wine has one poison immediately to the left.
The potions at either end are different, neither one is the ahead potion
The potions in positions 2 and 6 are the same (both wines or both poisons)
Two visually different potions do not contain poison (though we don't know their positions)
Knowledge of the position of the two non-poison potions gives enough information to deduce uniquely the positions of the ahead and back potions. (Hermione solved it.)

Using clues one two and three, we get the following eight possible sequences:

i	P	W	A	P	P	W	B
ii	P	W	P	A	P	W	B
iii

iv	B	P	A	P	W	P	W
v
vi
vii	B	P	W	A	P	P	W
viii	B	P	P	W	A	P	W
ix	B	P	W	P	A	P	W

x
xi
xii	P	P	W	A	B	P	W
xiii	P	P	W	B	A	P	W
xiv
xv

The next step is the tricky one; it's the application of clues 4 and 5 and a lot of people have trouble with this one. The reasoning goes like this:

Hermione solved this problem knowing the positions of the "dwarf" and the "giant" bottles. Obviously, if we had this information, the solution would be reasonably straightforward, but we unfortunatly, we don't. What we have to do is try to deduce their positions and when we do, we'll end up with some rather suprising results.

We begin by constructing a chart of all possible positions and compare this with our list of sequences we deduced from the first three clues. We immediately discover that some combination of dwarf and giant are impossible under the restrictions of the first three clues while others result in multiple sequences and a few result in a unique sequences.

		Position of Second Dwarf of Giant
		1	2	3	4	5	6
Position of First Dwarf or Giant	1
	2
	3	iv,vii,ix	i
	4	vii,viii	ii	vii,xii,xiii
	5	iv,vii-ix		iv,ix,xii,xiii	viii,xii,xiii
	6		i,ii	i	ii
	7	iv,vii-ix	i,ii	i,iv,vii,ix,xii,xiii	ii,vii,viii,xii,xiii	iv,viii,ix,xii,xiii	i,ii

The interpretation of this is that if the sequences listed for each combination of positions do not uniquely identify the positions of the ahead and behind potions (many thanks to Chris Calzonetti for pointing this out to me.), Hermione would not have been able to solve the puzzle.

Here's an example: Assume the dwarf and giant were in positions 1 and 5. The only sequences that didn't have poison in one of these position (applying clue 4) were: iv, vii, viii and ix. We lump viii and ix together since they have the same unique positions for the ahead potion and the back potion. This still leaves us with three possiblities:

iv	B	P	A	P	W	P	W
vii	B	P	W	A	P	P	W
viii,ix	B	P	.	.	A	P	W

Since we've used up all the available clues and still not found a unique solution, we have to conclude that the assumption of dwarf and giant in positions 1 and 5 is wrong.

Similarly, we can eliminate most combinations of dwarf and giant until we're left with four combinations:

Position of Dwarf and Giant	Resulting Pattern
2,3	i
3,6	i
2,4	ii
4,6	ii

Interestingly, the four possible positions of the dwarf and giant that result in a unique solution result in only two sequences and these sequences are related:

i	P	W	A	P	P	W	B
ii	P	W	P	A	P	W	B

We see that the ahead potion is always in position 3, 4 and that it's in either the dwarf or the giant bottle. The other (giant or dwarf) is wine and is in either position 2 or 6. The back potion is always in position 7.

Here's Hermione's solution:

"Got it," she said. "The smallest bottle will get us through the black fire -- towards the Stone."

Harry looked at the tiny bottle.

"There's only enough there for one of us," he said. "That's hardly one swallow."

They looked at each other.

"Which one will get you back through the purple flames?"

Hermione pointed at a rounded bottle at the right end of the line.

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Email: dave@tenletters.com