Simplified triple beam scale

Question

What could possibly be simpler than a triple beam scale?

^{(Ohaus Corp.)}

A simplified triple beam scale, of course:

1. 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.)

2. The beams’ weights need not be positive integers. (A helium balloon, for instance, can be a handy negative weight.)

3. The beams’ weights form an arithmetic progression, where the middle weight is simply the average of the other two weights.

4. 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.)

Answer 1

Because the description is a bit messy, I'll start with my interpretation:

1. On the right side, we have three different weights (can be negative) each individually switchable between positions 1 and 2, which we'll call "off" and "on" for convenience.
2. We can adjust the zero position (all three beams "off") of the scale, if we need to.
3. The weights on the right side need to be in linear progression
4. We can add a 1 gram weight on the left side when necessary

Were it not for point 3, we would just use (doubled) binary numbers: to make the numbers 0-7 by combining 1, 2, and 4 is trivial, so if we make the beams add 2g, 4g, and 8g, we can make all even numbers between 0 and 14. We then use the "zero adjust knob" to skew this by 2 grams (so that the "all weights off" situation needs 2 grams to balance), and we can make all even numbers from 2 to 16. From there we get to the odd weights by adding the 1g extra weight to the pan when necessary.

Symmetrically, we could also use -2g, -4g, and -8g as the weights, and adjust the zero so that "all beams off" is 16.

But point 3 is, of course, what this puzzle is all about. 2g, 4g, and 8g are not allowed, because 4g is closer to 2g than 8g; it needs to be in the middle. How can we make that work?

The answer seems to lie in

using negabinary numbers instead of binary numbers.

That neatly spaces the weights on the beams:

using -2g, 4g and -8g (or, symmetrically, 2g, -4g and 8g), we can still make eight consecutive even number weights, which we can "zero adjust" to coincide with 2-16.

This is exactly what we need, and -2 is smack in the middle of -8 and 4, so we have the required linear progression too.

To demonstrate that this works, let's pick the option with only one helium balloon. (Balloons are nice, but I hate subtraction.)

First, let's "zero adjust" so that "all beams off" needs 6g to balance. Then, we can measure like this:

 Desired weight | -4g |  2g |  8g | 1g in pan | zero adjust offset
 ---------------+-----+-----+-----+-----------+-------------------
           1g   | -4g |     |     | -1g       | +6g
           2g   | -4g |     |     |           | +6g
           3g   | -4g | +2g |     | -1g       | +6g
           4g   | -4g | +2g |     |           | +6g
           5g   |     |     |     | -1g       | +6g
           6g   |     |     |     |           | +6g
           7g   |     | +2g |     | -1g       | +6g
           8g   |     | +2g |     |           | +6g
           9g   | -4g |     | +8g | -1g       | +6g
          10g   | -4g |     | +8g |           | +6g
          11g   | -4g | +2g | +8g | -1g       | +6g
          12g   | -4g | +2g | +8g |           | +6g
          13g   |     |     | +8g | -1g       | +6g
          14g   |     |     | +8g |           | +6g
          15g   |     | +2g | +8g | -1g       | +6g
          16g   |     | +2g | +8g |           | +6g