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Puzzle:

You have 12 balls identical in size and appearance but one is an odd weight (it could be either light or heavy).

You have a set of scales which will give three possible readings: Left = Right, Left > Right or Left < Right (I.E Left and Right have equal weight, Left is Heavier, or Left is Lighter).

You
have only three chances to weigh the balls in any combination using the
scales. Determine which ball is the odd one and if it's heavier or
lighter than the rest. How do you do it?

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Solution

Number the balls 1, 2, 3,...10, 11, 12

Start off with them in three groups: [1, 2, 3 and 4], [5, 6, 7 and 8] and [9,10,11 and 12]

Weigh 1, 2, 3 and 4 vs. 5, 6, 7 and 8 with three possible outcomes:

1. If they balance, then 9,10,11,12 have the odd ball, so weigh 6,7,8 vs. 9,10,11 with three possible outcomes: 1a If 6,7,8 vs. 9,10,11 balances, 12 is the odd ball. Weigh it against any other ball to determine if it's heavy or light. 1b
If 9,10,11 is heavy, then they contain a heavy ball. Weigh 9 vs. 10. If
balanced, then 11 is the odd heavy ball, or else the heavier of 9 or 10
is the odd heavy ball. 1b If 9,10,11 is light, then they contain a
light ball. Weigh 9 vs. 10. If balanced, then 11 is the odd light ball,
or else the lighter of 9 or 10 is the odd light ball.

2. If
5,6,7,8 > 1,2,3,4 then either 5,6,7,8 contains a heavy ball or
1,2,3,4 contains a light ball so weigh 1,2,5 vs. 3,6,12 with three
possible outcomes: 2a If 1,2,5 vs. 3,6,12 balances, then either 4 is
the odd light ball or 7 or 8 is the odd heavy ball. Weigh 7 vs 8, if
they balance then 4 is the odd light ball, or the heaviest of 7 vs 8 is
the odd heavy ball. 2b If 3,6,12 is heavy, then either 6 is the odd
heavy ball or 1 or 2 is the odd light ball. Weigh 1 vs. 2. If balanced,
then 6 is the odd heavy ball, or else the lightest of 1 vs. 2 is the odd
light ball. 2c If 3,6,12 is light, then either 3 is light or 5 is
heavy. Weigh 3 against any other ball. If balanced, then 5 is the odd
heavy ball, or else 3 is the odd light ball.

3. If 1,2,3,4
> 5,6,7,8 then either 1,2,3,4 contains a heavy ball or 5,6,7,8
contains a light ball so weigh 5,6,1 vs. 7,2,12 with three possible
outcomes: 3a If 5,6,1 vs. 7,2,12 balances, then either 8 is the odd
light ball or 3 or 4 is the odd heavy ball. Weigh 3 vs 4.
If balanced, then 8 is the odd light ball, or else the heaviest of 3
vs. 4 is the odd heavy ball. 3b If 7,2,12 is heavy, then either 2 is
the odd heavy ball or 5 or 6 is the odd light ball. Weigh 5 vs. 6. If
balanced, then 2 is the odd heavy ball, or else the lightest of 5 vs. 6
is the odd light ball. 3c If 7,2,12 is light, then either 7 is light
or 1 is heavy. Weigh 7 against any other ball. If balanced, then 1 is
the odd heavy ball, or else 7 is the odd light ball.

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