Youtube channel Vsause2 recently posted a video on coin weighing problem. While the video is more or less an advertisement of their merchandise, the origin of the problem remains mysterious and catches my interest.
The problem reads: There are 50 rolls of pennies, each consisting of 50 pennies. Among them, 49 rolls are true pennies; each true penny weighs 2.5g. But one roll consists of 50 fake pennies, each fake penny weighs 2.4g. Use the spring scale once to figure out which roll is the fake roll.
I would like to know the origin of this problem.
Also consider the following variants:
(P2) You know that a true penny weighs 2.5g but don't know how much a fake penny weights (but you do know that fake pennies weigh the same). Then you need to use the spring scale twice.
(P3) you don't know how much pennies weigh (but true pennies weigh the same, and fake pennies weigh the same) Then you need to use the spring scale three times.
Essentially these are just linear algebra with integer coefficients ≤ 50. So I doubt that they had appeared in any official source (Martin Gardner's books, for instance). But I would like to hear if they had appeared before.
I don't know whether this was the original source, but it did turn up in Martin Gardner, Hexaflexagons and Other Mathematical Diversions, Chapter Three, Nine Problems, Problem 9, pp. 26-27, solutions pp. 35-36. This is the first of the variations you mention.