There is some literature on this problem. Lindström [2] and Cantor and Mills [1] independently give weighing strategies using 7 weighings, with the property that no weighing depends on the results of previous weighings. More generally, these papers prove that for every positive integer $k$, $2^{k}-1$ weighings suffice to determine which of $2^{k-1}k$ coins have which weights.

The weighing strategies are constructed recursively. The general process is a little involved, so I won't try to describe it here. If I understood everything correctly, the strategy given in [1] for 12 coins suggests putting the following coins on the scale:

- 1, 2, 4, 5, 7, 9, 11
- 2, 3, 5, 6, 12
- 1, 3, 5, 7, 8, 10
- 5, 6, 8, 9
- 1, 2, 4, 6, 8, 10
- 2, 3, 8, 9
- 1, 3, 6, 9


[1] Cantor, David G. and Mills, W.H.. Determination of a subset from certain combinatorial properties. *Canad. J. Math.* **18** 1966 42-48

[2] Lindström, Bernt. On a combinatorial problem in number theory. *Canad. Math. Bull.* **8** 1965 477-490