Hi,
I'm currently working on a little measurement/factorization/drafting problem, which can be summed up as 'how do you divide an established line into N uniform divisions without a measuring device'.
N may be up to 20.
Ideally, you can do this via a simple projection (mark out N same-size divisions on another line, then project lines to the ends of the target line, which establishes a vanishing point you can use to locate the other divisions on the target line.), or even a digital drawing tool that automatically marks divisions.
However, this isn't always practical, and it's preferable to be able to do things in a minimum of calculations / lines.
So I started working on this table, based whereever possible on divisions of 2 or 3, since they are easy to eyeball.
I'll post a few entries and explain the meaning, for a start:
* 4 : 2x2
* 5 : 2x2.5
So, to divide a line in 4, you divide it in 2 twice and you have the correct size for a segment. Easy to understand IMO.
Non-integer divisions are harder. When I write '2x2.5', I mean:
* first, divide the segment in 2, and consider one of the two sub sections you have made
* then, place a line that defines a unit such that 2.5 units fit into the subsection. One can start on this by first dividing it in half -- then you need to visualize how much to shrink the new half by so that 2.5 of it will fit into the subsection. This may seem a little fuzzy but you get the hang of it.
* If you placed the line correctly, you now have the correct unit size for dividing the line in 5.
Here's a gif showing the process:

Okay, going on with a few more entries from the table:
* 6 : 3x2
* 7 : 3x2.33...
* 8 : 2x2x2
* 9 : 3x3
Here, I'd further comment that I'm trying to limit the precision of non-integer divisions to no more precise than 1 in 4. This is because I don't really trust my eye to judge well measurements more precise than that. Here, the measurement for 7 is a third divided by 2 1/3.
* 10 : 3x3.33
* 11 : 3x3.66
* 12 : 3x2x2
Some of the more horrifying entries -- 3 1/3 is fairly hard to judge.
* 13 : 2x2x2x1.5
* 14 : 2x2x2x1.75
* 15 : 2x2x3.75 or 2x2x2.5x1.5
* 16 : 2x2x2x2
In comment on 15, my experiments suggest that smaller divisors are easier to cope with, so the 2x2x2.5x1.5 formula is actually easier than the 2x2x3.75 formula, IME. Probably because the relation is more gross and hence easier to judge.
* 17 : ??
* 18 : 2x2x2x2.25 or 3x2x2x1.5
* 19 : ??
* 20 : 2x2x2x2.5
This is where I'm currently stumped. I don't know of any divisors that are remotely nice for 17 or 19.
The best I have for 17 is 4.25x2x2 (which works, but judging the 4.25 size is pretty hard.)
19 is even worse -- the best I have is 4.75x2x2, which so far I've completely failed to estimate even once.
All other measures, I have successfully drawn using these formulas.
That's what I've got so far. If you have any suggestions on how to cope with 17, 19, or easier formulas for other numbers of divisions, I'd be interested to hear it.