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Optimal algorithm for finding the odd spheres
You have 12 balls and you know that they all weigh the same except for 1 which is heavier or lighter than all the others (you don't know which though). How can you make sure you know which ball is the heaviest/lightest in only 3 weighings?
The way I approached it was to split up the 12 balls into three sets of 4 and weigh two of the sets. If the sets balanced the scale, then I know the ball I am looking for must be in the set of 4 balls not weighed, else, I disregard said set and arbitrarily choose the heaviest set of 4 (as opposed to choosing the lightest set). I split the heaviest set of 4 balls into 2 and weigh that... etc.
Repeating this process until all 3 tries have been "used up", even if everything just so happened to be in your favor (the arbitrary choice you have in choosing the heaviest or lightest set happens to be the correct choice) in the end you still end up having to choose between 2 balls. A 50% chance is good, but I am wondering, is there a way to make sure 100%?