# Old School Measuring

You have an obscure job in 1600's Brussels. You make reference sticks. You own a high-quality ruler (which are quite rare in that era) and use it to cut lengths of wood or metal to certain lengths and send them out the various craftsmen who use them as a reference when performing their craft. For example, the King requested that all rapiers for his army's lieutenants be of a certain length, so you created a set of metal rods that were sent to dozens of different blacksmiths and scabbard makers so those rapiers and their sheaths are made (nearly) identically.

Your father was quite wealthy and you inherited his 4-foot ruler when he died, while your older brother got all of his money. The markings on the ruler are at every ligne, which are 144 to a foot. However, firearms are becoming increasingly popular and you know that craftsmen want more accurate reference sticks these days.

You take a trip to the master ruler-maker in the city (the greatest in the world) and inquire about making a finer ruler. He said he could make you a new ruler with any number of divisions you like, so long as they are equally spaced and no closer together than .8 ligne. He also notes that you, being far poorer than your father, only have enough money to purchase a ruler up to 12 ligne long.

You are able to use your current ruler and this new ruler in tandem, however you can't justify this purchase if you only improve your accuracy from 1 ligne to .8 ligne. Is there a way to get a better accuracy? The optimal answer has the highest accuracy, not the simpler method.

Rules:

A recent question about measuring ran into trouble with what was allowed in an answer, so I'll be explicit about what is allowed:

• You may treat the two rulers as having perfect markings. You can compare two markings (or ends of an object) and declare that one is “to the left” or “to the right” of the other. You can perfectly align two markings (or ends of an object) if you choose to do so.

• You are NOT a ruler-maker. If your solution involves transferring markings onto one the rulers or onto some other object, you cannot do this perfectly. Your transferred marking is no more accurate than your ability to measure it would be

• You don't own an angle-measuring tool (at least not an accurate one)

• The careful wording seems to imply that the length of the new ruler is fixed, but it doesn't quite say so anywhere. Can I choose any length for the new ruler, as long as it's 12 ligne or less?
– Bass
Jun 26 at 14:58
• @Bass yeah up to 12 ligne, just don't get the false impression you can get 12 1-ligne rulers. You're just buying one. Jun 26 at 16:27
• @Skosh Could you buy a 12-ligne ruler and cut it after every other number, giving you six 1-ligne rulers each with 2 markings? Jun 26 at 19:03
• @NuclearHoagie I guess I'd lean on rule 2 to say that your cuts cannot be more accurate than your ability to measure them. Jun 26 at 21:34

As opposed to codewarrior0's lazy approach, here's the toilsome way:

Have the new ruler divided into 8 parts, so that there are 7 markings between the two ends. But very importantly, request that (ostensibly, to save some money) the total length of the new ruler not be 12 ligne, but a bit shorter, namely

the diagonal of a $$8 \times 8$$ ligne square,

which handily keeps the line separation well over the 0.8 ligne threshold, at

$$\sqrt{2}$$ ligne, which is irrational with respect to the integer distances measurable by the other ruler, meaning that there is no integer $$m$$ whatsoever so that $$m\sqrt{2}$$ is an integer, which also means that $$m\sqrt{2} - \left\lfloor{m\sqrt{2}}\right\rfloor$$ (m of the new ruler units, but removing the integer part) is different for every $$m$$.

This allows you to measure

back and forth, counting an integer amount of units forward using your old ruler, and an irrational amount of units backwards using the new one, until you get "close enough" to the target distance,

which will eventually enable you to measure with the resolution of

"arbitrary precision" (or "precision within any given non-zero error limit") by establishing ever tighter upper and lower bounds for the measurement.

Of course this means your eyesight and the markings must be perfect, and that you don't make any errors whatsoever, all of which was explicitly deemed possible at the question's somewhat lengthy setup section.

The problem might arise with the final requirement of [not] unreasonable from the perspective of “a 17th century person needs to measure different length objects, dozens of times a day”, but that is of course easily remedied by only measuring to more lenient tolerances.

Here's a simulation (Try it Online!) of measuring a stick with a (previously unknown) length of Pi to the precision of one microligne using these two rulers.

• I don't understand how you implement this method within the lengths of the given rulers without making new marks of arbitrary precision on something (which is disallowed by the puzzle rules)
– fljx
Jun 27 at 7:00
• @fljx Align the marking on one ruler against the marking on the other, and keep count of the units you have measured in each direction.
– Bass
Jun 27 at 7:05
• @fljx No markings are required. You move the second ruler to align a pair of markings, then you move the first ruler to match a different pair of markings, then repeat. Jun 27 at 7:38
• Well I did say I'd be more critical of the ones that seemed complicated. There's a lot of ruler movement in this and I can't quite tell if you're taking advantage of something you shouldn't be able to. I would argue you can't have a point in the process where neither ruler is touching the measured object, and assuming that's a fair ruling I think you're going to end up with some physical space issues (I'm picturing the short ruler and the measured object want to be in the same location next to the big ruler) Jun 27 at 20:42
• @Skosh of course I'm taking advantage of everything you gave in the puzzle, doesn't seem likely that you can get infinite precision otherwise. :-) I don't suspect I'll be telling you how you could easily keep a ruler touching the object the entire time, because if the rules can change once, they'll surely keep changing until the suboptimal answer you had in mind becomes the only possible one. ;-)
– Bass
Jun 28 at 3:04