Why Your Ruler is Inefficient

We explore how it is possible to still measure every possible distance on a ruler, using fewer markings. We study a general solution to this problem in detail, and conclude by comparing our number of markings with a theoretical optimal lower bound.
The main idea behind our solution can be found in:
Alperin, R. C., & Drobot, V. (2011). Golomb rulers. Mathematics Magazine, 84(1), 48-55.
00:00 Intro
00:37 Application
01:55 First example: measuring 1-100
07:20 Why 10 is optimal
09:25 More general problem
11:46 k = qn case
14:15 k = qn + r case
16:59 Number of markings
20:55 How close to optimal is this?

[@guiperin6818]

bro i clicked to watch a funny story why the ruler could be better... and now i'm studing calculus again. Nice video

[@jsalsman]

Your facial expressions are really unique. You have a frequent look of deep concern which makes me feel like you care so much about whether people are understanding you, which in turn keeps me absolutely transfixed on your words. What a great attribute for an instructional KZbinr.

[@wirelessbaguette8997]

This is a very interesting problem conceptually, but in practice I do not believe fewer markings would be more efficient than maximal markings for general use! The mental energy required to find which combination of numbers work for what you want to measure would result in more energy used in measurement than the energy saved in marking production! There’s also the possibility of being unable to measure a distance with a ruler if there isn’t space to use the middle of the ruler for measurement, such as measuring distance from an interior corner. Essentially, this method is worth pursuing if what you are measuring is infrequent enough, or ruler manufacturing is so expensive (like interferometry), that the energy savings on fewer markings is > energy spent on the more complicated measuring process.

[@user-sl6gn1ss8p]

he gives the example of arrays of satellites - there it makes way more sense. So you can have, say, 7 satellites, but 15 different inter-satellite distances with equal spacing. I don't think the idea is ever to actually apply this for regular rulers.

[@BRLN1]

​@user-sl6gn1ss8p although it doesn't make sense to be applied to rulers in general, but it sounds like a funny gimmick, e.g. in Vsauce's curiosity box 😅

[@joevano]

Don’t get stuck on a ruler… it’s just a convenient way to visualize the concept. He concedes right at the beginning that this isn’t super practical for a real ruler.

[@Starnite_]

there is in machining size blocks you can use, you generally want to use as few of these blocks(they are a determined size such as one 1cm and then one 2 cm and so forth, and to measure your tools you can use this stratagy to use as few as possible of these blocks to get the final length needed for example

[@joevano]

@@Starnite_ same thing with weights and a scale

[@islandfireballkill]

This is really interesting. I wonder what kind of solution a greedy algorithm would be able to get. A really simple one being choose 0, k then the least number that gives you the most new distances.

[@DrBarker]

This may give rise to a ruler marked more like the example at the start. We could mark 0, k, then a mark at 1 allows us to measure 1 and k - 1. Next we can mark 3, which allows us to measure 2, 3, and k - 3. The next marking would be 7, which allows us to measure 4, 6, 7, and k - 7. Depending how big k is, we could continue like this, with the next marking allowing us to measure 5 new distances. I can imagine it would be quite difficult to deal with the remaining missing lengths, like how we can't measure 5 yet in this example.

[@Patrik6920]

@@DrBarker well.. for most uses it wouldent be very usefull... and it would complicate things alot for childeren, especially for kids with learning disabilities or nerophysological disorders.. ...but for scientiffic reasons yes we already uses log scales so.. ..and usin noiner scales, so .. the only thing i would see it would do is to complicate things for ppl that dont need things to be complicated... a tape measurer with 0, 1, 3, 6, 10, 15, 21, 31, 42, 55 ..and so on marked would be very hard to use inutively... u need lines anyway for measuring fractions...

[@severinghams]

​@@Patrik6920 Nobody is saying that its going to replace the standard ruler, this is obviously a novel concept; its completely useless in tight spaces and only works with whole numbers. Also, I should well hope you're not implying we order society to lower the bar all the way down to the lowest common denominator, so that "no child is left behind" because instead of making sure kids who can become super smart become super smart while identifying kids who need more help so that they can get that help, we would actually leave every kid behind at the same time, because instead of just the dumb kids falling behind, you're making sure none of the kids get a proper education. Real cool, man. Smart kids need smart stuff. Things like this teaches children about the relationship between numbers. Dumb kids need special help understanding things, so that they fan function just as well as smart kids. And nobody is saying either is more important than the other; God loves each and every one of us equally. I'm telling you that there is no playing field thats exactly level. Never. Some people are better at some things than others. We absolutely should not lower the entry level bar for what children should know at the age they are at. Exceptions are for specifically learning disabled children. Letting the smart kids have smart stuff, or even just trying to teach all the kids about smart stuff wont make the dumb kids explode. It won't kill them, it wont make them sick, and it wont turn them upside down and shake the money out of their pockets. In fact, it will help them, because then the teacher will know who to explain things better to, and who to help.

[@severinghams]

​@@Patrik6920 Nobody is saying that its going to replace the standard ruler, this is obviously a novel concept; its completely useless in tight spaces and only works with whole numbers. Also, I should well hope you're not implying we order society to lower the bar all the way down to the lowest common denominator, so that "no child is left behind" because instead of making sure kids who can become super smart become super smart while identifying kids who need more help so that they can get that help, we would actually leave every kid behind at the same time, because instead of just the less intelligent kids falling behind, you're making sure none of the kids get a proper education. Real cool, man. Smart kids need smart stuff. Things like this teaches children about the relationship between numbers. Less intelligent kids need special help understanding things, so that they can function just as well as the smart kids. And nobody is saying either is more important than the other; God loves each and every one of us equally. What I'm actually telling you that there is no playing field thats exactly level. Never. Some people are better at some things than others. But we absolutely should never lower the entry level bar for what children should know at the age they are at. Exceptions are for specifically learning disabled children. Letting the smart kids have smart stuff, or even just trying to teach all the kids about smart stuff wont make the special needs kids explode. It won't kill them, it wont make them sick, and it wont turn them upside down and shake the money out of their pockets. In fact, it will help them, because then the teacher will know who to explain things better to, and who to help.

[@severinghams]

​@@Patrik6920 Nobody is saying that its going to replace the standard ruler, this is obviously a novel concept; its completely useless in tight spaces and only works with whole numbers. Also, I should well hope you're not implying we order society to lower the bar all the way down to the lowest common denominator, so that "no child is left behind" because instead of making sure kids who can become super smart become super smart while identifying kids who need more help so that they can get that help, we would actually leave every kid behind at the same time, because instead of just the less intelligent kids falling behind, you're making sure none of the kids get a proper education. Real cool, man. Smart kids need smart stuff. Things like this teaches children about the relationship between numbers. Less intelligent kids need special help understanding things, so that they can function just as well as the smart kids. And nobody is saying either is more important than the other; God loves each and every one of us equally. What I'm actually telling you that there is no playing field thats exactly level. Never. Some people are better at some things than others. But we absolutely should never lower the entry level bar for what children should know at the age they are at. Exceptions are for specifically learning disabled children. Letting the smart kids have smart stuff, or even just trying to teach all the kids about smart stuff wont make the special needs kids explode. It won't kill them, it wont make them sick, and it wont turn them upside down and shake the money out of their pockets. In fact, it will help them, because then the teacher will know who to explain things better to, and who to help.

[@mrigayu]

This reminds me of trying to find a basis with the least number of vectors possible that will still span a vector space. This may be really naïve, but I wonder if there’s a way to cast the problem in the framework of linear algebra and if that could even help. Anyways, this was a great exploration! It’s a problem that isn’t immediately apparent, at least to me, but with a few hints along the way it’s quite nice what comes out!

[@DrBarker]

If we look at the span of a vector space, we would expect to be able to add/subtract different multiples of the basis elements. I suppose this could correspond to having multiple rulers, e.g. if you can measure 3 on a ruler, then you can measure 6 using 2 rulers, or 9 using 3 rulers, etc.

[@awaredeshmukh3202]

There are an astonishing number of people in these comments who don't recognize a mathematical puzzle when they see one, and are complaining about how this isn't actually very efficient. Why would someone watch this if they weren't actually curious about the math?

[@warmpianist]

I had this almost exact question when I was applying for intern during undergrad, except you are only allowed to add numbers, not subtract (must use two rulers to get from 0 to N). There is a similar visualization of using Cantor set and recursively solve for the best solution. Divide 0 to N into 3 equal sections, and remove the second section. For the two other sections, recursively divide and remove again until the section has a size of 1. I will comment further if you have any questions.

[@warmpianist]

Interviewer didn't explicitly give me solution but only told me to do a trial and error. Also, I was asked to solve for general case with more than 2 rulers. The Cantor set also applies similarly.

[@vnXun]

"Why Your Ruler is Inefficient?" Probably because he "inherited" the power from his dad and made sure he remain in charge by rigging the elec... oh you mean a physical ruler, my bad

[@ElchiKing]

I'm just running a brute force search to see how high we can go for a fixed number of markings. So far, I found e.g. the sequence (0, 1, 3, 6, 13, 20, 27, 31, 35, 36) which gets up to 36 with 10 marks (the check whether this is the maximum is not yet done). This is 2 marks better than the 12 using your method, but still 1 worse than the hypothetical optimum of 9.

[@jeffeloso]

Same for multiple tappings on a transformer to give a range of voltage options.

[@random19911004]

This seems to be somewhat similar to another problem I recently saw. See the video: "Egg Dropping Puzzle with 2 Eggs and 100 Floors'"

[@DrBarker]

This is an interesting connection! There's some of the same sort of structure, especially dropping the egg from floors in intervals of 10, which gives an answer of 19 drops needed, like how we needed 19 markings on our ruler.

[@filedotjar]

For finding the number of markings, i played around on desmos and realixed that you could do (sqrt(a) * 2) - 1. It seems quite similar to your algorithm, but rather than getting an upper bound, it seems to just result in the exact value for the cases I tried. If I'm missing something, feel free to correct me!

[@DrBarker]

I was able to get an upper bound using the method in the video of 2sqrt(k), rather than 2sqrt(k) + 2, but this involved quite a lot of fiddly algebra, so I left this out from the video. I can imagine this upper bound could be improved a bit more, without having to adapt the method of his to mark the ruler.

[@cmmp5510]

Very nice. I will use it. I'll make a ruler to my workshop with your system and try it.

[@Zephei]

This problem reminds me of difference sets from combinatorics. Given a group G of order v, a (v,k,λ) difference set of G is a subset D of size k such that every non-identity element of G can be written as a difference ab^(-1) of elements a,b in D in exactly λ distinct ways. If our ruler of length n(n-1)+1 was in the form of a loop, optimal solutions to the problem in the video would correspond to (n(n-1)+1,n,1) difference sets of the cyclic group of order n(n-1)+1. If n-1 is a prime power, then such sets are known to exist, and it has been an open problem since the 1930s whether difference sets with λ=1 exist for any other finite abelian groups (the conjecture is that none exist, and this has been verified for n ≤ 2,000,000 in a paper by Daniel M. Gordon).

[@BRLN1]

You brought up a good example with the spacing of large radio antennas. Lets stick to this example: Assume (due to budget constraints or whatever other reason) you know we will build 5 this year, another 5 next year, 10 more the year after and so on. Now each year we want to use and hence space out the available antennas in an optimal way, such that we can always continue to add in the subsequent year. In other words the ruler gets longer and longer, but we are only allowed to add new markers. We cannot change or delete existing markers. Do you have any idea how this will influence the solution?

[@mikefochtman7164]

An interesting problem. ISTR there is a Very Large Array of radio telescopes that are mobile, so the spacing can be reconfigured. This is certainly more costly per antenna, but wonderfully flexible. But a question I have is how they deal with earth's rotation? Sure, the mounting allows them to pan/tilt each antenna, but the spacing cannot be changed that fast. So when viewed from the radio source, the spacing elongates as the array moves with the earth over the horizon. Must be some 'maths' involved ;)

[@ytkerfuffles6429]

i havent finished the video but i believe the 19 one can be optimized further if you replace the 20-100 with the optimal 0-8 solution

[@KaiseruSoze]

How would you verify the length of a meter stick?

[@mollaconan]

Measuring weight (ie., mass) using a balance scale version of it: What is the minimum (or efficient) set of weights in grams to weigh masses, say, 1 to 100 grams? (1, 3, 9, 27, 81)

[@pedroavellarcosta9389]

there a numberphile on that, I believe. pratical numbers if you look for

[@pfeilspitze]

Balanced ternary is best base.

[@protocol6]

Can you optimize a trundle/surveyor/measuring wheel to go with your optimal ruler? Basically, the same but with a finite ring or field? For any given pair of marks, there's now two possible measurements, either the positive difference or the modulus minus the negative difference.

[@curtiswfranks]

This was good. I will need to review, but it is fascinating.

[@maxxieway26]

Excellent video! I would be interested to know how one would determine which values for K allow for a perfect set of markings such that each difference can only be measured with 1 pair of markings. I wonder if there are infinitely many rulers that have this property!

[@JamesSmith-ix5jd]

Generally you can pack more information into less space/symbols, but it comes at a cost of longer time of unpacking in your mind afterwards, the good example are APL programming language or regex or things like that, if you ever delt with some densely packed information you know how pita it is to unpack in your mind while reading or working with it.

[@apv]

Since n is actually an integer, I wonder if there is a discrete optimization technique that can be used to kind the optimal n rather than expanding it to the reals to do calculus.

[@emanuellandeholm5657]

Reminds me of the McNugget numbers.

[@samueldeandrade8535]

Oh my Euler, that's genius.

[@Utesfan100]

It seems like most of the redundant numbers come from trying to get all the integers near the high end of the ruler. It might be interesting to ask what is the smallest ruler with n tick marks having n(n+1)/2 distinct integer distances. For 1, 2 and 3 the answer seems obvious. (tick marks at 0 1 and 3 respectively) For 4 I found one with a length of 7, and for 5 one with a length of 13 and all lengths through 9. What is the smallest ruler with unique distances having the first n numbers?

[@lukewatson8848]

If you're comfortable with moving your ruler, you can get away with only having one marking on every ruler!

[@gblargg]

6:00 That seems like the most practical. Have markings of 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, ... 10 ... 20 ... up to whatever. Then measure from the right, aligning to a number, then with the left edge on the left side between 10 and 0 to get the ones.

[@roger7341]

My ruler can't add or subtract, but it can multiply and divide. How is it marked off?

[@MrRyanroberson1]

0,1,3,6,10 is the triangular sequence so i can imagine there is a methid that first assumes all the triangular numners and then fills in the gaps. For this case: 0 1 3 6 10 15 has gaps 1 2 3 4 5 6 7 9 10 12 14 15 with holes 8 11 13. The holes have gaps 3, 2 which means we need a signature gap sequence of 3 and then 2 somewhere in the existing sequence, forward or backward, which occurs only in 6 3 1, with the contrasting number being 14 (it must be on the right here, but negatives are not forbidden in my book since they just represent a repositioning of the 0 marker). This proves to me that in the triangular number method this must be the optimal solution for 15

[@MrRyanroberson1]

Consider instead a more intelligent expanding series which skips existing gaps first: 0 1 3 7 12 20 30 44 missing 15 16 17 22 25 26 28 31 33 34 35 36 38 39 40 42 Adding 59 fills in 15 29 39 47 52 56 58 59 each time always adding totally unique entries. You can stop at any point and switch strategies to fill in the blanks.

[@MrRyanroberson1]

I wrote a longer reply but youtube delered it. Suffice to say i found 6 markings for 1-12 and 8 for 1-20 with 0 1 3 7 12 and 20 as the base with 11 and 17 18 as supporting marks respectively.

[@mikefochtman7164]

@ 14:00 for 'case 1', it seems a trivial proof to show that q must be equal to n, right?? After all n=floor(sqrt(k)) and if n is a factor of k, it must be n=sqrt(k) and thus q= sqrt(k).

[@STEAMerBear]

This reminds me of my own pretend country’s currency problem in which students are asked how to print the fewest bill denominations to carry any integer amount of money efficiently (in a fairly compact form). Instead of solving that, I’ll leave it for you to chew on.

[@user-ik2kd9mb5t]

You can't do 15 for "theoretical lower bound". There are 15 pairs between 6 marks and there always will be an integer between 1 and 15 without pair of marks of this length on the ruler. But you can do 13 for 6: [0, 1, 2, 6, 10, 13]

[@Fire_Axus]

11:00 we need to find the integer minimum for the optimal number

[@AJoe-ze6go]

Inefficient in what sense? Fewer markings make for a more efficient manufacturing process; but that ruler requires more cognitive load because you have to do math on the fly whenever you use it. Which do you think generally happens more - the number of times a ruler is made, or the number of times it's used?

[@mikeguitar9769]

So Zeno’s ruler was theoretically “efficient” but he made it “inefficient” by trying to measure all the way to the end.

[@gbpferrao]

What about milimiters

[@pudinsarudin]

Binary ruler?

[@danielrhouck]

Your 15-length example in the beginning is not based on the scheme in the rest of the video, and in fact does better. Is it just the Strong Law of Small Numbers that it appears to be based on triangle numbers? I doubt it because those are also O(√k) and they are closely related to the k choose 2 in your lower bound, but if there were a simple scheme based on them I’d expect you to have explained that instead.

[@Pystro]

I can't believe someone else came up with this same idea. I felt quite stupid when I thought of this a few years ago.

[@Pystro]

After watching it, it's interesting how you approached the solution from a different direction. I was trying to find sets of lines that wouldn't duplicate distances, while hitting as many of the small distances consecutively as possible. And I just worked out a few small examples instead of arriving at that general square root strategy. 0,1 (1 possible distance) 0,1,3 or 0,2,3 (3 possible distances) 0,1,4,6 or 0,2,5,6 (6 possible distances) 0,3,4,9,11 or 0,2,7,8,11 (10 possible distances, but gets 11 instead of 10) And 6 lines with 15 possible distances can't hit all lowest 14 integers (I'm 99.9% sure).

[@bozhidarmihaylov]

This is a “thumb Up” in YT’s algorithm binary based measurement system :)

[@wesleydeng71]

Sparse rulers.

[@lenskihe]

Sparse rulers rule!

[@bob8776]

Is this why every person I hire says they can “read a measuring tape” but act like they’re doing calculus when I ask for a simple measurement?

[@poorgrammar3136]

Is this NP? If so that is interesting

[@Hosib-st1hd]

"Draw a 7cm long line" FUCK

[@matthewlyons6544]

While this is all well and good if you want to cut something a certain length, and need to mark that interval, it becomes far less useful if you're trying to measure something with unknown length.

[@casualTetrisFan]

these are called Golomb rulers, right? (I discovered this name through Murderous Maths, I think?)

[@Pystro]

Technically, these are "sparse rulers" (which have the condition to be able to measure all distances up to their length. "Golomb rulers" have the condition to not duplicate any distances.

[@awaredeshmukh3202]

Murderous Maths!! I haven't met anyone else who's read them. I don't think I ever read that one though.

[@gammahits9727]

mm, cm and dm. (the rulers)

[@ronindebeatrice]

Do the English really use a 15cm ruler? Does it make them feel larger?

[@smzig]

That's all good an well if you only measure in integers. Unfortunately, real life requires measuring fractional lengths as well. Granted, I know the main part of this video is the mathematics behind the fewest markings on an integer ruler and not necessarily trying to reinvent the ruler in general.

[@bobh6728]

@smzig It isn’t even practical if you know your object is of integer length. Say you have tape measure 30 meters long with the minimum number of markings. You want to measure an object that about 15 meters long. You would have to line up one end with the first mark and then go the other end and see if it lands on an another mark or not. If not, you start at the second mark and repeat over and over until you get the correct measure. It would be interesting to figure out the most efficient way to measure with this type of ruler. I’m sure it would take longer than a normal one and would outweigh any savings in not adding the extra lines. The telescopes he mentioned would be an actual efficient use.

[@ProHolmes]

​@@bobh6728 agree. I think for measuring objects such ruler in inpractical. But for drawing a segments of a chosen integer length, well maybe it can work pretty well, but I don't think it would be faster than a traditional ruler.

[@GhostyOcean]

9:00 if you're going to use calculus, might as weel say it is a minimum because y''(10)>0, so it is concave up.

[@KSignalEingang]

I would have assumed the answer would involve logarithms somehow, this seems like a problem that would be tailor-made for them.

[@pfeilspitze]

Look up "radix economy", which does use logarithms to show that 3 is the best (integer) base.

[@guru0503p]

❤

[@blacklight683]

You can also just have 1m to count any length

[@CatNolara]

This kind of ruler would make it really easy to mess up and read it wrong. Also if you want to measure the distance from e.g. the edge of a room you would be screwed, because you'd have to stick the ruler through the wall to get to the number you need. Guess falls under the category of solutions to problems that don't exist.

[@57thorns]

0, 1, 5, 9, 12, 14 and 15 are equivalent with the 0,1,3,6,10,14,15 one.

[@Leivoso]

A better title would've been "A math puzzle about rulers" or something like that, because I'm pretty sure the normal ruler is just better at measuring than a ruler with a complicated method. Don't get me wrong, it's an awesome math problem, but in real life a normal ruler would just be the easier way most of the time

[@avaraportti1873]

Turns out I only needed a ruler that can measure 9 cm

[@MrPathakhemant]

The New type of ruler needs extra mental efforts ans skill to add/subtract numbers from/to the required number and hence it is not practically useful for all.

[@Nikioko]

A slide rule has logarithmic markings...

[@dliessmgg]

can't wait until you encounter a ruler that shows millimeters

[@davidroddini1512]

The people I work with can't even read a ruler with all the markings, let alone a ruler that's missing some percentage of the markings.

[@biggerdoofus]

This is a good example of why the human concept of "efficiency" is flawed. The problem of measurement has multiple competing separate aspects where that word could be applied, and mutually exclusively. Sure, having the fewest markings for the most _integers_ is one way to be "efficient", but most rulers meant for actual use are not restricted to integers (or not using the same unit scale if you want to think of it that way). As such, a tape measurer is the preferred "efficient" version, in that it's physically efficient in use of space while maximizing the actual use cases. Meanwhile the telescope example is "efficient" in a space and monetary sense, while still having other ways to maximize usefulness.

[@Pystro]

"most rulers meant for actual use are not restricted to integers" _Actually,_ rulers typically have a minimal line separation (1mm or 1/16th of an inch for example) and any distances they can measure are integer multiples of that. But in reality it's also important that for _every_ measurement, there's a line a fraction of a unit before the object's length and one a fraction of a unit after, since that makes it possible to estimate a more accurate measurement to below the line separation.

[@bobbun9630]

This ruler wouldn't have been useful in a lot of my science classes back in the day, as I was supposed to be estimating one decimal place between markings. The problem as stated obviously assumes all distances are natural number distances, and with that in mind penny pinching ruler manufacturers who want to save ink by offloading work onto their customers can actually get by with even less ink. I don't need any set of marks to measure out a distance of three explicitly, for example, provided that there are marks that will allow both two and four to be measured This follows from the only natural number greater than two and less than four being three. Accounting for this sort of implicit measurement should eliminate many marks, even if it does likely ruin some of the other applications of the technique.

[@STEAMerBear]

This is clever!

[@xXJ4FARGAMERXx]

That was a long way to say: I measure decimals, and this works only on non-decimals. Sure it's useful for the companies, but it isn't useful to me.

[@bobbun9630]

@@xXJ4FARGAMERXx If that's your conclusion, you should have read more carefully.

[@ARichli]

Unnecessary propagation of errors?

[@1RandomToaster]

That sounds all great and mathy but if I knew how long the thing I wanted to measure was I wouldn’t need to measure it.

[@awaredeshmukh3202]

That's equally valid for regular rulers though. You wouldn't try to measure your walls with a foot long ruler, because you know you're going to need something more like a tape measure, etc

[@rubenvasquez8592]

Efficiency is not oriented towards the instrument in itself but towards the application. a) if I want to make a series of mesurements of equal length on a material, it is easier to lay down the ruler and follow the multiples of the meassure, instead of placing and lifting the ruler for each measurment. b) a ruler of this design can only be used in the specific segment with the specific measurement. If I want to measure something bounded on one side, my measurement will be limited with how far the ruler can go in. c) in practical terms it takes more effort to teach how to use this design sinse it is not very intuitive and the marking are not self explanatory.

[@vukkulvar9769]

b) Imagine if you needed to measure 4cm from the wall using the 15cm ruler graduated (0,1,3,6,10,14,15).

[@martinr7728]

He already said at the very beginning of the video that we use regular markings because it's easy. This is meant to be a fun mathematical problem, with applications in other areas. "Efficiency" in the sense of how many markings you theoretically need, not how many are practical.

[@omerelhagahmed551]

n= floor(k) OR CEIL(k) Just a reminder, because it's not written on the board

[@tylerbakeman]

Lets compare points on a ruler to telescopes, and not something more similar. The only scientists that should be allowed to tell us how to measure lengths efficiently, are the different types of geometrists. Obviously, if you use a ruler/ tape measurer, it is 99% of the time better to have the standard setup, because you have random access to any sub-length in the ruler. It’s a tool that works exactly the way it should- because it works. Theres nothing wrong with alternative rulers- if they work. But yeah. Standard rulers are great. They should go metric in the US

[@nealjroberts4050]

Technically it's only inefficient if markers, and probably rulers, are expensive to make and apply, and the time spent to calculate is irrelevant.

[@vukkulvar9769]

And then you end up needing to measure 4cm from the wall using the 15cm ruler graduated (0,1,3,6,10,14,15).

[@IroAppe]

Oh dear, it seems that many people only read the title, and did not even watch 40 SECONDS into the video.

[@KSignalEingang]

They only watched the first ten seconds, then skipped ahead and watched the 20th second, then the 30th, etc, and assumed they could interpolate the rest.

[@DrBarker]

@@KSignalEingang 😂😂😂

[@McGhinch]

Well, I know many people who would be clueless how to measure fractions -- and it is time consuming. Your "meter-example" makes this obvious. You have to cut 76 centimeters from an unspecified length of material. With a ruler giving me all centimeter fractions from 1 to 100 this is just one measurement. To make it that simple, you only need fewer markings: 0, 1, 2, 4, 8, 16, 32, 64, 100 to measure any full centimeter distance between 0 and 100.

[@usernamename2978]

All done without distracting music, inane grins or stupid jokes. If this doesn't cause the KZbin servers to disintegrate, nothing will.

[@Jerry10939]

You can do all that but it’s more work. I want to measure 86. I want to measure directly to 86 and not measure to 50 or 100 and measure back because it can get confusing. I think your way is inefficient.

[@abdulazizalzamil7612]

why isnt 0,1,3,6,10,15,21,28,36,45,55,66,78,91,100 the optimal solution it only has 15 markings.

[@Eye-vp5de]

Impossible to measure 98, probably some other numbers too (probably 51)

[@Fire_Axus]

hint: 100/n + n - 1

[@Fire_Axus]

another hint: sqrt(n)

[@larrystuder6378]

All those markings make it easier for not-as-smart people- we don't have to DO arithmetic, just know our numbers...

[@ProHolmes]

Also it's faster to just see all markins, instead of spending time to spot a proper interval between this upgraded markings. Also common ruler do have mm markings, and they are needed most of the times if we talk about a ruler not a 1m+ measuting tape. So we need mm markings, not all figures but all makings. 15 cm ruler with 1cm precision is pretty useless..

[@Nihil2407]

Why don't you just use a binary scale? A marking at 0, 1, 2, 4, 8, 16... It isn't hard to see, that all measurements are in here and the efficiency should be somewhere around ld(k). On the ruler you used, the number of markings grows linearly, on the binary ruler it's logarithmic, so actually it would be an order of magnitude

[@Eye-vp5de]

I have had the same question, but he answers it in the end (there simply aren't enough combinations of pairs of markings to have all integers between 1 and n) Example: 64 0 1 2 4 8 16 32 64 Impossible to measure 61,59 and many other numbers

[@eleghari]

Leave my ruler alone! 😶

[@leechesg]

This premise is unfortunately defeated by the average person's unwillingness to think that hard when they just want to measure twelve damn centimeters already. "Measure from 3 to 15? why not just put a 12 on it?" Would be a common complaint, and I can't say i would blame anybody for it. Sometimes simplicity is more efficient than efficiency, when you consider the human element.

[@joseislanio8910]

Also, the efficiency is lost once you have to measure an unknown distance and you have to try multiple combinations until you get it right

[@leechesg]

@@joseislanio8910 Somehow I didn't even consider that, but now that you mentioned it, just the idea of moving the ruler back and forth at least three times is annoying me. And considering that, it would also be ineffective wherever there's limited space; if you can only barely fit the ruler into a gap, then you can't accurately measure certain distances at all within that gap

[@hexagon8899]

now try measuring something near a wall

[@Pystro]

If you're allowed to make a mark, you can always go out the full length of the ruler, and then come back the rest. But there's probably better rulers for that set of rules. Even the "trivial ruler" of length 1 could do that (though not in 2 steps).

[@hexagon8899]

@@Pystro alright then, between 2 walls just over 1 ruler length apart

[@DanBurgaud]

2:50 Carpenters are already busy with all the work and you want them to count backwards from 100 to 48 to get 52?? VERY BAD IDEA!

[@ExistenceUniversity]

I see you have never actually used a ruler for real life lol

[@Andrumen01]

Like when you need to measure 13cm exactly...but there's no 13cm. You gotta remember that most abstractions that mathematicians do are for very intelligent people...I suggest you search for George Carlin and see what his opinion is on the intelligence of the average person. Also, have you ever taught undergrad courses? Tells you a lot about why ideas that look like good ideas are actually bad ideas....

[@Eye-vp5de]

There is 13 between 1 and 14. You're wrong.

[@user-xk9dl8xw8l]

Sparse rulers are mathematically interesting, but that's all

[@jamesstrickland833]

You can even be more efficient and just use the powers of 3. Using 1, 3, 9, etc. Will allow you to get every positive integer

[@Eye-vp5de]

No, check his argument for the optimal solution 0,1,3,9,27 Can't measure 25

[@kw91]

Not an engineer confirmed

[@realdragon]

With interferometry yeah kinda, but still having more telescopes in between where "you don't need them" wouldn't hurt

[@awaredeshmukh3202]

What the astronomers wouldn't give for More Telescopes... unfortunately getting governments to give them money is not always easy

[@realdragon]

@@awaredeshmukh3202 We should stop spending more money on everything and invest in telescopes. 98% of taxes should go into building telescopes

[@awaredeshmukh3202]

@@realdragon absolutely. Telescopes are the one true human good, nothing else matters in the face of massive photon collecting power

[@user-rcghjewqw]

Dr Backer could name his video like "Optimal placing of telescopes" and stop polluting informational space.

[@deadman7770]

yapping level 999

[@G4M3_0V3R]

I hate how the letter K is written this way. It's R!

[@rotten-Z]

How can all this nonsense be more effective than one simple measurement? Especially when you are paid for the work done, and not for the number of measurements?

[@99temporal]

Did you even watch the video? He used it to explain how for applications(like astronomy) where each numbers on the ruler would be analogous to building a whole other very expensive measuring device, the second one would be way more efficient

[@ProHolmes]

​@@99temporalso how telescope distribution in astronomy makes existence of creatively marked 15-30 cm plain rules more meanful. Probably taking a ruler as a subject to this was not the best choice.

[@Antuan2911]

Come on! That silly and crazy... LoL~! I need all the marks in my ruler... When I want to measure two points in the paper I need all the marks and all subdivisions for that... And don''t tell me to do it in 2 or 3 steps, cause that will cause accuracy problems.

[@bart2019]

But now you're forcing people to calculate the difference between 2 markings, and quite a lot of people would really not like that. Not everybody is into math.

[@honounome]

god forbid doing 14 - 6

[@hexagon8899]

@@honounome it has been forbidden!

[@yxlxfxf]

the problem is not the subtraction itself, it's having to solve "find 2 numbers in the set whose difference is X" problem every time I need to use the ruler. Gets even more annoying for 30cm rules.

[@sebastianmanterfield3132]

​@@yxlxfxfthis only takes linear time, you can check for each number if that number + n is on the ruler

[@yxlxfxf]

@@sebastianmanterfield3132 checking if a number is on the ruler is not constant time. It can be O(log N) I guess since the numbers are sorted in the ruler and you can do binary search. Either way, my point is that measuring with a standard ruler is O(1) time always. And yes, I know if you were to model this problem inside a computer you would use a hash set or something similar for O(1) lookups, but in practice all you have is a ruler (aka list of sorted numbers)

[@SzymekCSC]

WRONG

[@kaltziferYT]

Cause he is really old and also corrupted kgb agent from soviet era. From Russia with love.

[@MuhammadyusufK]

A033437