# Solution to "Twelve Coins"

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 11 - - - - - - - - - - -
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

CoinTypePattern
1heavyLLB
2heavyLLR
3heavyLBL
4lightRBR
5lightRBL
6lightRLR
7heavyLRB
8lightRLL
9heavyBRR
10lightBRB
11lightBRL
12heavyBBR

The corresponding coin positions are

OperationLeftRightNot used
Weighing 11 2 3 74 5 6 89 10 11 12
Weighing 21 2 6 87 9 10 113 4 5 12
Weighing 33 5 8 112 6 9 121 4 7 10

Which is a possible solution.

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