I was sorting out the coins in my loose-change jar the other day, and the following thought crossed my head: Is it possible to deduce the number of each type of coin in this pile by simply weighing them?
The coins I were counting were Australian 5c, 10c, 20c and 50c. According to Wikipedia, they weights are:
5c 2.83 grams 10c 5.65 grams 20c 11.30 grams 50c 15.55 grams
But this might be a particularly bad choice of weights for the coins. Since 565 divides 1130, we can't tell the difference (through weighing) between two 10c coins and one 20c coin. It seems that two 5c coins and one 10c coin would be hard to distinguish also.
So my question is:
Question: What would be a better way to designate the weights of these coins so that we could (in most cases) uniquely determine the number of each type of coin in a single weighing?
This would be subject to some practical constraints:
- Each coin is a reasonably light, but not too light (e.g. between 2 and 20 grams).
- If A and B are two multisets of coins, and A and B have equal weights, then |A| and |B| should be very large (more than is likely to be in a typical jar).
- The scales do not measure with infinite precision. Coins are not minted with infinitely accurate weights.
- The weights of the coins must differ by a reasonable amount (e.g. by at least 3 grams).