There are many different solutions to this puzzle.
This solution is elegant in that the pattern of the weighings does not
depend on the results of the weighings.

Each weighing can have 3 possible results:

Left Side Heavy (L)

Level (B)

Right Side Heavy (R)

This gives 3 × 3 × 3 = 27 possible results, although all three level
(denoted BBB ) is not useful. If we number the coins 1 to 12 then there
are 24 possible answers,

1 light

1 heavy

2 light

2 heavy, etc.

Let us arbitrarily decide that we will always put 4 coins in each pan
with 4 coins not being weighed and further that

Weighing 1 Left side heavy

Weighing 2 Left side heavy

Weighing 3 Balance (denoted LLB) shall mean that coin 1 is heavy

Then it follows that, with the same coins in each pan, RRL will imply that coin 1
is light. Further, it follows that coin 1 must have been in the Left pan for
weighings 1 & 2 and in neither pan for weighing 3.

Operation | Left Pan | Right pan | Not used |
---|---|---|---|

Weighing 1 | 1 - - - | - - - - | - - - - |

Weighing 2 | 1 - - - | - - - - | - - - - |

Weighing 3 | - - - - | - - - - | 1 - - - |

We can continue assigning answers to results, while placing coins in the pans as required and eventually arrive at a solution similar to

Coin | Type | Pattern |
---|---|---|

1 | heavy | LLB |

2 | heavy | LLR |

3 | heavy | LBL |

4 | light | RBR |

5 | light | RBL |

6 | light | RLR |

7 | heavy | LRB |

8 | light | RLL |

9 | heavy | BRR |

10 | light | BRB |

11 | light | BRL |

12 | heavy | BBR |

The corresponding coin positions are

Operation | Left | Right | Not used |
---|---|---|---|

Weighing 1 | 1 2 3 7 | 4 5 6 8 | 9 10 11 12 |

Weighing 2 | 1 2 6 8 | 7 9 10 11 | 3 4 5 12 |

Weighing 3 | 3 5 8 11 | 2 6 9 12 | 1 4 7 10 |

Which is a possible solution.

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