Binary in the shop

[mklotz]

I wanted to make some extension rods for one of my dial indicators - simple brass rods with a male 4-48 thread at one end and a female 4-48 thread at the other end. The question is: What lengths to make the rods?

Since it's a one inch DI, I'd like to be able to assemble the rods in different configurations to cover all the integral one inch lengths, i.e. 1,2,3,... inches.

The immediate thought is make rods in lengths 1,2,3 inches. Then I can construct every combination up to 6 inches...

1 = 1
2 = 2
3 = 3
4 = 3+1
5 = 3+2
6 = 3+2+1

But then it occurred to me that, for the same construction effort, i.e. making three rods, I could get one more inch of extension with the combination 1,2,4

1 = 1
2 = 2
3 = 1+2
4 = 4
5 = 4+1
6 = 4+2
7 = 4+2+1

People with computer exposure will instantly recognize this as a binary progression. I believe, but haven't yet been able to rigorously prove, that a binary progression will always give you the longest range for the least work in any problem that falls into this category. [Making weights for a balance scale would be another classic example.]

The binary effect arises from the fact that an extension rod or a balance weight can either be "in" the final product or "not". So, if you want to construct all the integers (up to some maximum) your individual pieces all have to have "weights" corresponding to the powers of two less than the maximum.

An interesting side effect is that you can easily determine which rods/weights you need for a final number by simply writing the desired final number in binary.