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12 Coin problem

You have 12 coins that appear identical. However, one is counterfeit and may may either lighter or heavier than the other eleven coins. Determine the conterfeit coin and whether it is light or heavy in three weighings using a balance scale.

Solution

Arbitarily label the coins A, B, C, D, E, F, G, H, I, J, K, L.

First weighing

Left: A,B,C,D.    Right: E,F,G,H.

There are three possible outcomes:-

  1. Case 1 : Equality
    This means that the counterfeit coin is one of I, J, K or L, but you don't know whether it is heavy or light.
  2. Case 2 : Left side heavier
    This can mean one of two possibilities:-
    a) The counterfeit coin is one of A, B, C or D and it is heavier, or,
    b) The counterfeit coin is one of E, F, G or H and it is lighter.
  3. Case 3 : Right side heavier
    This is just a mirror of the above situation. The principal of the solution is the same.
Second weighing

Case 1    Left: I,J,K    Right: A,B,C (known "good" coins)

There are three possible outcomes:-

  1. Case 1a : Equality
    This means that coin L is counterfeit, but you don't know whether it is heavy or light.
  2. Case 1b : Left side heavier
    The counterfeit coin is one of I, J or K and it is heavier.
  3. Case 1c : Right side heavier
    The counterfeit coin is one of I, J or K and it is lighter.
Case 2    Left: A,B,C,E    Right: D,I,J,K
                Note: A,B,C,D may be heavy; E may be light and I,J,K are "good".

There are three possible outcomes:-

  1. Case 2a : Equality
    The counterfeit coin is one of F, G or H and it is lighter.
  2. Case 2b : Left side heavier
    The counterfeit coin is one of A, B or C and it is heavier.
  3. Case 2c : Right side heavier
    This can mean one of two possibilities:-
    a) The counterfeit coin is D and it is heavier, or,
    b) The counterfeit coin is E and it is lighter.
Third weighing

There are now four different senarios to resolve:-

1)    Counterfeit coin known, but not whether it is heavy or light (case 1a)

Simply compare the coin with any of the others.

2)    Counterfeit coin is either (say) X and heavy or (say) Y and light (case 2c)

Compare X with a "good" coin.
If X is heavier then that's your coin, else it's Y.

3)    Counterfeit coin is heavy and one of three (say X,Y,Z) (cases 1b & 2b)

Compare X with Y
If X heavier, then X is the coin.
If Y heavier, then Y is the coin.
If equal, then Z is the coin.

4)    Counterfeit coin is light and one of three (say X,Y,Z) (cases 1c & 2a)

Same principal as for senario 3).

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