What could possibly be simpler than a triple beam scale?
A simplified triple beam scale, of course:
Just one identically spaced pair of notches on each beam, allowing 2 possible weight positions per beam for a total of 2 × 2 × 2 = 8 possible combinations between all three beams. (No gradually sliding weights. No attachment weights to hang onto the balance arm.)
The beams’ weights need not be positive integers. (A helium balloon, for instance, can be a handy negative weight.)
The beams’ weights form an arithmetic progression, where the middle weight is simply the average of the other two weights.
An additional 1-unit free weight is available for calibration but may also be added to the weighing pan during measurement.
Such a scale does have a perfectly useful application — to measure out 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 or 16 units of sample in a single weighing, . . .
. . . with what selection of weights on the three beams?
(Notes: Two essentially equivalent solutions. No need to be able to measure 0 units of sample. The distance between each arm’s notches equals the pan’s lever arm, so the arms’ weights are as heavy as the amounts of sample they balance. What are called arms’ weights here are also known as poises and riders.)