13 coins weighing problem

Divide the coins into three sets of four coins each. For a first weight trial place a group of four coins in each pan. There are two possibilities, which we shall investigate separately:

(1) The pans balance.
(2) One pan outweighs the other.

WHEN THE PANS BALANCE.
In the event the counterfeit is in the unweighed set, all eight coins on the scale are genuine. Number the coins in the doubtful group 1, 2, 3, 4. Carry out a second weight trial by placing coins 1, 2 and 3 in one pan and placing three genuine coins in the other pan. There are two possibilities:

(A) The pans balance. In this case, coin 4 is counterfeit. A third weighing, comparing coin 4 with a genuine one, will tell whether it is lighter or heavier.

(B) One pan is heavier. In this case, the counterfeit is one of the coins 1, 2 or 3. If the genuine coins are heavier, then the counterfeit is a light coin, and vice versa. One more weight trial will identify which of coins 1, 2 or 3 is counterfeit. (1 in one pan, 2 in the other; if they balance, then 3 is counterfeit; if one side is heavy and it was determined that the counterfeit is heavy, then that is your coin; if it was determined that the counterfeit is light, then the other coin is your coin.)

WHEN ONE PAN OUTWEIGHS THE OTHER.
In this case, all other coins are genuine. Designate the coins in the heavy pan as 1, 2, 3, 4 (if one of these coins is false, then it is heavier than the others) and the coins in the lighter pan by 1', 2', 3', 4' (if one of these coins is false, then it is lighter than the others). A second weight trial will be made by placing coins 1, 2, 1' in one pan and coins 3, 4, 2' in the other. Again, there are several possibilities:

(A) The pans balance. In this case, the counterfeit coin is either 3' or 4' (and is lighter than a genuine coin). A third weight trial is made by placing coin 3' in one pan and coin 4' in the other; the lighter coin will be counterfeit.

(B) The pan containing coins 1, 2 and 1' is heavier. In this case, coins 3, 4 and 1' are genuine; were either coin 3 or coin 4 heavier than the other, or were coin 1' light, then in the second weight trial the pan containing coins 3, 4 and 2' would have been heavy, which was not the result in this case. Therefore, the counterfeit coin is either coin 1 or 2 (and it is a heavier coin), or else it is coin 2' (and it is a lighter coin). A third weight trial is made, placing coin 1 in one pan and coin 2 in the other. If the pans balance, then the counterfeit coin is 2'. If the pans fail to balance, then the counterfeit coin is in the heavier pan.

(C) The pan containing coins 3, 4 and 2' is heavier. Reasoning as before, we conclude that coins 1, 2 and 2' are genuine and that if coin 3 or 4 is counterfeit, then it is a heavier coin than the others, and if coin 1' is the counterfeit, then it is lighter. A third weighing is made by placing coin 3 in one pan and coin 4 in the other. If the pans balance, then the counterfeit is 1'; if, on the other hand, one pan is heavier, then it contains the counterfeit coin.

The 12 balls weighing problem Solution