12 ball problem

The problem is you have 12 balls (of any kind), one ball either
lighter or heavier than the other 11. A scale is provided to you but only
three weighs are allowed. How can you possibly figure out which is the odd
ball in three iterations.

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Easy answer is as follows. it takes max 4 iterations for this:

take 6 balls on one scale and other 6 on second scale. (1)
the scales will not balance out.
take the heavier 6
weight 3 against 3 (2)
if they balance out then the ball is lighter one and is found in the other 6
balance 3 - 3 from the first set of 6 which was lighter on the scale.(3)
take the 3 balls which goes up (lighter)
balance 1 ball on left and one ball on right, keep the third ball on table. (4)
if they balance out : ball on table is the lighter ball
else, the scale which goes up has the lighter ball.
else if they do not balance out
the culprit is a heavier one.
take the 3 balls which is heavier and balance 1 against another, keep third on
table (3)
if they balance out : ball on table is the lighter ball
else, the scale which goes up has the lighter ball.


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the efficient solution of finding the ball in 3 iterations is as follows
answer from: Games for the Super-
intelligent" by the late Jim Fixx.

Number the balls 1 to 12. Weigh 1, 2, 3, and 4 against 5, 6, 7, and 8.
If (1, 2, 3, 4) and (5, 6, 7, 8) balance:
Weigh 9 and 10 against 11 and 8 (we know 8 is not the odd ball).
If (9, 10) and (11, 8) balance: then 12 is the odd one.

Weigh 12 against any other to find out if it is heavy or light.

If (9, 10) and (11, 8) do not balance: suppose 11 and 8 are heavier,
than 9 and 10; then either 11 is heavy, or 9 is light, or 10 is light.

Weigh 9 against 10; if they balance, 11 is heavy; if they do not,
the lighter of 9 and 10 is the odd ball.

(Similar argument if 11 and 8 are lighter than 9 and 10).

If (1, 2, 3, 4) and (5, 6, 7, 8) do not balance:
Suppose 5, 6, 7, and 8 are heavier than 1, 2, 3, & 4. Then: one of
(1, 2, 3, or 4) is light, or else one of (5, 6, 7, or 8) is heavy.
Weigh 1, 2, and 5 against 3, 6, and 9.
If they balance: then either 7 is heavy, or 8 is heavy, or 4 is light.
Weigh 7 against 8; if they balance, 4 is the odd ball, otherwise the
heavier of 7 and 8 is the odd ball.

If (1, 2, 5) and (3, 6, 9) do not balance: suppose 1, 2, and 5 are lighter
than 3, 6, and 9; then either 6 is heavy, or 1 is light, or 2 is light.
Weigh 1 against 2 to find out which one of the three choices is true.
Otherwise, suppose 1, 2, and 5 are heavier than 3, 6, and 9; then either 3
is light, or 5 is heavy.

Weigh 3 against (say) 2 to find out which of the two choices is true.

(Similar argument if 1, 2, and 5 are lighter than 3, 6, and 9).