IBM Research | Ponder This | November 2002 solutions

# Ponder This

## November 2002

<<October November December>>

Although we didn't realize it at the time, this month's puzzle was due to Martin Gardner, in the February 1979 issue of Scientific American.

Five moves.

We will denote positions of bottles as "U", "D" or "?", for "up", "down" or "unknown", respectively.
We list the four positions clockwise, up to unknown rotation. Initially they are "? ? ? ?".

First move: Take two opposite bottles and turn them "up".
Now their setup is "? U ? U".

Second move: Take two adjacent bottles and turn them "up".
Now the setup is "? U U U".
Assuming the referee does not stop the game, we know in fact that the setup is "D U U U".

Third move: Take two opposite bottles.
If one is "down", turn it "up", so that we win ("U U U U").
Otherwise they are both "up"; turn one "down", so that we are left with "D D U U".

Fourth move: Take two adjacent bottles and turn them over.
If they were both "up", we win with "D D D D".
If they were both "down", we win with "U U U U".
Otherwise our setup becomes "D U D U".

Fifth move: Take two opposite bottles and turn them over, winning.

Why is four moves impossible?

There are four non-winning positions (up to rotation):
P1 = "U U U D",
P2 = "D D D U",
P3 = "U D U D",
P4 = "U U D D".
Suppose for the moment that we even know which position we are in.
From P3 it requires one move to win.
From P4 it requires two moves to guarantee a win;
one move does not suffice, since whether we pick "opposite" or "adjacent"
the two untouched bottles can have differing orientations.
From P1 it requires three moves to guarantee a win:
if we pick "opposite" we can be given two "up" bottles, and can
only get to P2 or P1 or P4;
if we pick "adjacent" we can be given two "up" bottles (and not
know which ones), and no matter what we do, the referee can
force us into P1 or P2 or P4.

Return to the real game, in which we don't know whether we are starting with P1, P2, P3 or P4.
On our first move, whether we select "adjacent" or "opposite", the referee will arrange for the two bottles we see to have differing orientations (one "up", one "down"), and no matter what we do at that point, we will be left with a set of possibilities that contains one of the sets {P1,P3} or {P1,P4} or {P2,P3} or {P2,P4}.
(For example, if we choose "adjacent" and then change "UD" to "UU" we will be left with either P1 or P4, and not know which.)
On our second move, no matter what we do, we will be left with a set of possibilities that includes P1 (or else it includes P2).
Then even if the referee tells us our position, we still require three more moves to finish.

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