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Solution to the 12 ball weighing brain teaster

Suppose one ball among twelve identical balls has a different weight. Using a balance scale, how can you find this ball and tell whether it is heavier or lighter, using only three weighings.

Solution

Weigh 4 against 4 (assume one side goes up or down)

Next remove 2 from one side and 1 from the other side. This leaves 3 against 2 balls, so add one normal weight ball to the side with 2 balls.

Now switch around one ball from each side and weigh. If the scale tilts the opposite way then the unknown ball is among the two balls that switched place. If the scale tilts the same way then the unknown ball is among the three original balls that did not shift place, If the scale balances then the unknown ball is among the three that were removed from original set of 8.

What I have described is the most complex case of reducing a set of 8 to a set of 3 unknowns in two weighings. There are a lot of things I haven't gone over, but they are all simpler cases derived from the above scenario.

Now here is how to find the odd ball from a set of three in one weighing. Since we know whether the balls could be heavy or light from the first weighing, it is easy to bring it down to one. For example, consider if 2 balls weighed the scale down and the 3'rd ball rose on the scale. This means that either one of the two balls are heavy or the third ball is light. Now simply weight the two suspected heavy ones together. Your ball will be the one that weighs the scale down. If they balance then the ball is a light one, the third one that was not weighed.

Try and solve the problem of reducing 4 unkown balls to 1 known in three weighing. This is not too dificult.