math_fanatic
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The minimum number of weighings is 3, but if we don't know if the counterfeit coin is heavier or lighter, you'd have to get lucky on your second weighing, otherwise, it's going to take 4 weighings to find the counterfeit.
Step #1: First you weigh 6 of the coins against the other 6. One side is going to be lighter than the other because the counterfeit is either heavier or lighter than the other 11 coins.
At this point you have to make a guess as to whether the counterfeit is lighter or heavier. The process will be the same whether you guess that it's heavier or lighter.
Step #2:
Option 1: you decide that the counterfeit is lighter. Put aside the heavier group of 6 then take the 6 coins from the lighter group and divide them into groups of 3. Weigh the groups of 3 against each other. If the groups of 3 don't weigh the same, you've determined that the counterfeit is lighter so you can go directly to step #4 which will be your final weighing (the minimum number of 3).
If the two groups of 3 weigh the same, you have determined that the counterfeit is heavier and you must proceed to step #3 which will cost you a 4th weighing.
Option 2: you decide that the counterfeit is heavier. Put aside the lighter group of 6 then take the 6 coins from the heavier group and divide them into groups of 3. Weigh the groups of 3 against each other. If the groups don't weigh the same, you've determined that the counterfeit is heavier so you can go directly to step #4 which will be your final weighing (the minimum number of 3).
If the two groups of 3 weigh the same, you have determined that the counterfeit is lighter and you must proceed to step #3 which will cost you a 4th weighing.
Step #3: Divide the group of 6 that you put aside after your first weighing and divide them into groups of 3. Weigh the groups against each other.
If you have determined that the counterfeit is heavier, put aside the group of 3 that is lighter (since you now know that the counterfeit is heavier, it must be within the heavier group).
If you have determined that the counterfeit is lighter, put aside the group of 3 that is heavier (since you now know that the counterfeit is lighter, it must be within the lighter group).
Step#4: Take the final group of 3 coins (the lighter group if the counterfeit is lighter or the heavy group if the counterfeit is heavier) and put one aside. Weigh two of the coins against each other.
If one of the coins on the scale is heavier than the other and you've determined that the counterfeit is lighter, then the lighter coin is the counterfeit. If the two coins weigh the same, the coin you put aside is the counterfeit.
If one of the coins on the scale is heavier than the other and you've determined that the counterfeit is heavier, then the heavier coin is the counterfeit. If the two coins weigh the same, the coin you put aside is the counterfeit.
 Posted 159 day ago
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