Length of an elastic band around 3 circles

[@pizzatc0165]

Finally a problem I was able to solve

[@gilangalghozali7275]

hahahha.. same

[@jikaikas]

Lol same

[@charly6052]

Loool me too

[@doclee8755]

No you didn’t. It’s okay though. Try next time!

[@charly6052]

But i have a probleme solve for C: Tan(c).tan(4c)=-1

[@Anonymous-lc5qu]

If you do this with 4 circles touching each another, length will be 4+π. For 5 circles 5+π. For 6, 6+π and so on.

[@39rama]

I was thinking exactly the same thing and got the same pattern. One complaint I have about Presh's videos is that a problem is often not generalized when it can be. If time is a constraint, at least ask the general question and let the viewers think about it.

[@102728]

Or 2nr+2πr with n being the amount of circles. Easier to read would be replacing 2r with d, to form the equation nd+πd with n>=2

[@rafiqhaq]

6 circles, 6+π; 5 circles, 5+π; 4 circles, 4+π; 3 circles, 3+π; 2 circles, 2+π; 1 circle? 0 circles?

[@102728]

@@rafiqhaq it breaks down under 2 circles, which is why i added the requirement

[@rafiqhaq]

But why does it break down and not follow the pattern?

[@NoizyInSeattle]

I like problems where it turns out: oh, that was easier than I thought.

[@BlacksmithTWD]

Didn't you pause the video to come up with your own solution before watching the rest?

[@fillmainlol5973]

@@BlacksmithTWD if you needed to pause, ur slow at maths sorry i don’t make the rules

[@tuyiren781]

@@fillmainlol5973 if you didn't pause and solved in 2 seconds, you are lying, sorry I don't make the rules

[@davidloh3147]

@@tuyiren781 i did pause to do the math, but i could juat look at it and see that that makes sense as a solution and then got it doing da math

[@uzairkhanpathan7563]

@@fillmainlol5973 :joy:

[@itayshtainberg7408]

My thinking method was that the band has 3 curves with "equal curvature" each, and the total curvature in degrees has to be equal to the curvature of a circle (otherwise the band would not close in a loop). Therefore each curve segment is equal to a third of a circle

[@invincibleflesh4526]

Same, but I thought of the slope at each point of disconnect as being why it must be a full circle, rather than its being a closed loop.

[@goki6548]

Exactly.

[@tapiocaweasel]

my thoughts exactly

[@prajwalchoudhary4824]

Amazing

[@ataberk6845]

Yeah same

[@SlidellRobotics]

I worked this several years ago as a subset of a bigger problem - a bundle of N circles [actually cylinders, and particularly lengths of PVC pipe]. Because I was actually trying to figure out best storage for tubes which could be built into cubicles with a door, my real target numbers were 12 and 23. The best answers I found for these two were hex grids, 2+3+4+3 and 2+3+4+5+4+3+2, later "confirmed" with elastic and actual pieces of pipe.

[@royjonesrampage6684]

I think its 3+pi. Really intuitive tbh since the collision area on each circle is 1/3 of the circle(120 degrees as opposed to 180 with 2 circles and a band or 360 with 1 circle or 90 with 4 circles) and the non collision areas are each equal to 2 half circle lengths each aka 1/3pi*3+3*0,5*2=pi+3

[@shekharr2227]

2π+3 right? Nevertheless, this is exactly how I solved

[@ps.2]

It's even simpler - the curved bits have to add up to a full circle (since, you know, the path is a full loop) no matter what the individual arc angles are.

[@MErcuryOnVenuS]

d=1,r= ½

[@ps.2]

In fact this problem is a nice illustration of that principle that if you take a closed convex shape (e.g., Presh's interior triangle) and draw a larger shape a uniform distance _x_ away, the new perimeter is τx larger than the old. The form this usually takes: if you stretch a rope around a circumference of the Earth, then raise the rope by 1 meter, how much rope must you add? Of course the answer is τ meters.

[@csababekesi-marton2393]

@@shekharr2227 No. The circumference of the circle (or we should say 3 * 1/3 circles) = d*π = 1*π. So the final solution will be 3+π.

[@TonyCrenshawsLatte]

At a first glance, I went "oooh that's gonna be a tough one." Then I started thinking about how I would solve it, and then I realized this was actually a pretty simple problem, and that I could reason it out all in my head! This was a feel-good puzzle for me. :D

[@GnosisArizona]

This is by far, one of the most useful channels on KZbin; educational and entertaining! Big Thank You!

[@anandk9220]

(3 + pi) units is my answer. Just tried it orally and enjoyed solving this. Actually it's very easy. The only thing to be careful is to find central angle at the band along the points of contact of one circle.

[@anandk9220]

@@yaroslavpanych2067 I realised that after watching video solution. Although I knew those tangent-radii will help form 2 right angles at center apart from 60°, I imagined little differently. Obviously those 3 arcs joined together will make circle which is to be realised.

[@srinitaaigaura]

That's easy. It's a straight triangle, so it can only ever be 180-60 = 120 degrees, or one third the cirumference.

[@daffa_fm4583]

this is the first problem i actually tried to solve: length of band touching 1 circle: pi/3 length of band not touching a circle: 1*3 = 3 length of band: 3(pi/3) + 3 = pi + 3

[@popogast]

(3+pi)*1 which is the same.

[@紫衣-j3b]

en.. too simple this one

[@leif1075]

Excuse me you're just guessing arent you theres no evidence length of band touching one circle is pi/3m

[@daffa_fm4583]

@@leif1075 extend the lines of the straight parts of the band, you can see that it forms an equilateral triangle, now draw 2 lines connecting one of the circles center to the parts of the triangle thats touching the circle, you can see the angle at the center is 120 degrees and that means the length of band touching one circle (120 degrees arc) is pi/3

[@紫衣-j3b]

@@leif1075 from any point on the band, along with the band, go back to the point, is it a 360 degree? no matter how many circles, try other numbers, 4,5,6...? so all curved portions = a circle.

[@rangerbeast]

The comfort u feel after solving this channel's problem

[@randomdude9135]

Ikr ☺

[@safrprojects]

Intuitively, because of tangent lines, it should be 3*diameter + 1*circumference

[@Jivvi]

That's what I was thinking, as soon as I saw it. The "explanation" made it seem so much more complicated than it needed to be.

[@Ibakecookiess]

@@Jivvi The explanation follows the same idea but is an actual proof.

[@M_J_9_7]

*2:35* Nope,... he's wrong *Length of band ≤ 3+π* Bcoz it's an elastic band and it's already stretched 😎

[@pe3akpe3et99]

finally found a correct solution

[@M_J_9_7]

@@pe3akpe3et99 Overthink to solve simple problems - Presh He's the one who taught us and he.....

[@pianistg_7893]

No wayyy this physic guy destroyed our mathmatician minds ): Btw it made me laugh a lot xd

[@M_J_9_7]

@@pianistg_7893 he should have just said it a string.... No probs

[@marc_frank]

also the length of the band should be taken from it's center, so it'll be longer, since it has thickness

[@christopherbrosz5003]

This is an interesting problem with practical application. A machinist may use similar theory to define a tool path to machine parts as well as the speeds and feeds of the cut. This can all be outputted from a CAM software, but can be calculated manually for simple geometry.

[@pranavdharme5630]

Second part we can also solve using the Arc length formula which arc length S=r*theta so here r = 120° = 2π/3 and then add all 3 arc length to get 3*2π/3 = 2π and here r = 1/2 so S = (2π)*(1/2) = π. So we get our ans = 3+π

[@armwrestling_nerd]

Spontaneously : π +3 (πD + 3 ) because rounded parts are 360 °(to form the loop) so that's a full circle circumference and straight line sections are the same as D...

[@ExplosiveBrohoof]

My solution: The band has two classes of parts to it: the curvy parts and the straight parts, and there are three of each. The curvy parts each are equal to 1/3 of the circle's circumference, so in total the length of the curvy parts is 3(pi/3) = pi. Let AB be a line segment representing one of the three straight part of the band, so that one of the endpoints is A and one of the endpoints is B. Let A' be the center of the circle that A lies on, and define B' similarly. Because AB is tangent to both of these circles, AA' is perpendicular to AB, as is BB'. Hence, ABB'A' is a rectangle, and AB and A'B' have the same length. A'B' has length 1, since it is the line segment connecting the center of two externally tangent circles each of radius 1/2. Thus, AB has length 1, and so the total length of the straight parts is 3. Therefore the total length of the band is 3 + pi.

[@blockyhour4224]

I thought I'd have to do the thing where a line is tangent to 2 distant circles and passes through the center but the fact that i don't even remember the name probably says enough Edit: i was talking about internal/external tangents

[@Allangulon]

Secant line.

[@blockyhour4224]

@@Allangulon ye that but with 2 circles where the centers of the circle and line are collinear

[@leif1075]

@@blockyhour4224 what is that and how would that even help??

[@blockyhour4224]

@@leif1075 idek if it helps i just thought it'd be a step to the solution

[@gustavgnoettgen]

I thought that the curves were more complicated. Good to know for other cases!

[@egilsandnes9637]

Usually the MindYourDecitions problems are too hard for me to solve. This time I glanced at the thumbnail and ... well nailed it. You might say I thumbnailed it.

[@niteshsharma6803]

Direct formula :- (nd+2πr) ❤️ where (n= no. Of circle in triangular shape and touch rubber band too) (d = diameter)

[@deepjyoti5610]

U have to tell orientation of circle, U can't calculate length of total elastic for any oreintation case with this formula Take 3 circle in straight line (R=1/2)then calculate length of elastic length with thiss u will get 3+π, but check it will 4+π , This formula only work for several cases

[@niteshsharma6803]

@@deepjyoti5610 circles which touch each another in a triangular shape... for the example.. put 4 circle then put 3 on 4 then 2 and 1.... pyramid shape in 2D form.. I hope u understand...but make sure there are total 10 circles but value of n will be 9 bcoz we include that circles which touch rubber band

[@deepjyoti5610]

@@niteshsharma6803 yes for all pyramidal cases it works

[@niteshsharma6803]

@@deepjyoti5610 yes.. 😍🥳

[@V1DE0DR0ME]

Thanks! Great general formula. You might want to edit to say n = no. of circle that touches the rubber band (not each other)

[@BDCOMBO]

Spent three days doing this by finding lines points and tangents only to figure out the delicious little trick at the end, loved this one

[@bergamt]

“It looks like pi + 3, but I’m sure there’s some subtle reason it’s more complicated than that” [watches video] “Nope”

[@pro-nav]

When you don't use pen and paper 😂

[@Zorlig]

Yeah, easy!

[@mawavoy]

@JoelBenstein, trust your instinct. Then try to verify by asking and answering the question why does my hunch seem correct. The verification will be the solution. I was stumped for a while as to how to measure the straight line segments, so I worked with two range t circles. There the length of the side straight lines was easy to see.

[@randoscience4756]

Ikr ahahaha

[@tomalcolm]

Big brain!

[@glitchedspoon]

This is the only video where I have genuinely worked out before the solution was revealed.

[@TheMonkeyGrape]

love this one looks so tough, but is so elegant

[@rich_in_paradise]

So I figured out in my head that it was 3+pi, but for the curved sections I felt it was OK to just state they added up to one circumference because the band in total has to go around 360 degrees of turn to get back to the start (the shape is convex). I wonder if that argument would hold muster on an exam though.

[@heh2393]

It's perfectly valid

[@LaeeqKhan01]

How will your argument hold if circles had different radii?

[@rich_in_paradise]

@@LaeeqKhan01 It wouldn't. The shape would still be convex, but if the radii are different there would be a different portion of that angle on each circle. And since the radii are different you need to know both to calculate the length. Using the method shown in the video you could calculate it, after using trigonometry to calculate the angles at each corner of the triangle that connects the centres of the circles.

[@geoninja8971]

Awesome.... finally I solve one exactly the same way as Presh! :)

[@Xploiter]

hi

[@anandskanth6326]

Me too

[@raghavr2601]

Same here !

[@Rafa-rt3ri]

Me too

[@nigarnazarli1711]

Me too)

[@doclee8755]

What a beautiful use of reasoning. So many of today’s young people have very poor logic skills. Doing lots of training like this is a great way to sharpen your reasoning and math skills. This was a great problem! Thanks.

[@warpromo6636]

there's also a lot of young people with good logic skills, it's shown that every 30 years iq increases by 9 points, a child is more likely to grow fundamentally smarter than you when they're at your age

[@warpromo6636]

also these math videos don't make your logic better, although they may get you into the thinking mood

[@Rohit_03]

It was easy but i calculated length of arc separately and didnt thought they would make a circle.

[@smanzoli]

It´s correct and nice to note it´s number of spheres + PI... works with any positive integer number

[@nibaranghosh2202]

Mind blowing! 😊 Go ahead. May you live long and god bless you. 😇

[@JrDarkPhantom]

I was able to solve this in my head just from the image on the thumbnail before even clicking on the video. Then I just forwarded to the very end to check my answer :) I was correct.

[@zivssps]

You can generalize the question. For n >=2, the length will be pi + n.

[@26enaksheemondal82]

I don't think that's going to work(I tried), if it could, could you explain me sir/madam?

[@gabiold]

@@26enaksheemondal82 D(pi + n) to be precise. There is n straight segments of D lengths, and as you increase the number of circles each curved section has less angle, exactly 360deg/n, so they always add up to a whole circumfence eventually.

[@WowOafus]

The first one I was able to solve as soon as I saw the thumbnail. Nice to have an easier one once in a while. Thanks

[@gamefacierglitches]

Taught myself something new today about shortest distance around _n_ circles of equal radii

[@bendystrawz2832]

Underrated comment. Maybe he'll do a bonus video about it. He usually explores the generic versions of these problems where applicable. Although, I guess it would be a really short video, so maybe not, lol.

[@namehere2986]

circles with the same radii ofcourse ... also is it n+ (d*pi)? just a random guess prolly wrong

[@gamefacierglitches]

@@namehere2986 yes same radii, I should add that. Edit: Just realized a small phrasing error One thing to note is the equation only uses convex hull circles, excluding any other circles. So for a touching adjacent hull arrangement like this, its (n*d) + (d*pi). d happened to be 1 so it became n + (d*pi). For any other arrangement, it would be (d*pi) + (total distance between the center of each adjacent hull along the perimeter). And just thought of the formula for different radii circles, but it's a bit complex so I need to write it down first.

[@namehere2986]

@@gamefacierglitches i can alr tel ur smart with all the mathematical jargon I had to google (never heard of a convex hull.. My brain is too hull for this... Get it?...)lmao.. I just guessed that formula trying to think of a possible corelation without actually checking (guess it proves im lazy) . Also the radii wld be 0.5 for the circles cuz the diameter is 1.

[@gamefacierglitches]

@@namehere2986 lol. I meant d not r (oopsies).

[@phasm42]

You can also think of it in terms of traveling the perimeter. You only turn on the curved portions, and you must complete a full 360 in one trip around the perimeter. If you remove all the white portions of the diagram, the three circular sections will come together to form a circle.

[@Bry10022]

Very interesting problem and answer…

[@tatsutakamaro]

Я минут за 20 решил, спросони. Только угол 120 * иначе нашёл - через достраивание внешнего треугольника и сумму углов ромба. Но так даже элегантнее.

[@yzakvargas4968]

I found it to be scalable for circles with diameters more than 1: Length of band = D(3+ π)

[@tiltil9442]

Same, same (not different).

[@anshumanagrawal346]

@@tiltil9442 lol what?

[@mwb911]

and I believe further generalized with the number (n) of circles: D(n + π)

[@yzakvargas4968]

@@mwb911 , darn you beat me to it, haha! Yes, you're right, elegant, isn't it?

[@randomsandwichian]

Yes, I'd have also calculated it as total of diameters + total of ⅓ of each circumference.

[@guyedwards22]

I used symmetry reasoning to figure out the arcs corresponding to pi/3: the picture is three-fold rotation symmetric, and points on the circle tangential to the rubber band map to each other under those rotations. 2Pi/3 is the angle of rotation for such symmetry, and the actual arc length is half of that since the radius is 1/2.

[@prakharyadav1909]

3 + π I solved it in the thumbnail. Good explanation though.

[@narada6329]

Same

[@indigoziona]

I got Alice, Bob and Charlie, I was close on the 25 mechanical horses, but I needed to stop halfway through the solution of this before I could figure it out!

[@haraldusterud2965]

As the question is "length of band" and it is elastic, I would say it is just about a handful of units. In order for it to remain on the cylinders, it will necessarily need to be shorter than the actual circumference (pi + 3). You do not measure elastic bands in the stretched condition...

[@allylilith5605]

at the moment where it's in use (which we consider) it's length is 3+pi

[@RajaAnbazhagan]

You must be fun at parties.

[@steveshadforth]

Take your meds

[@RudeGuyGames]

There's pedantry, and then there's this.

[@douglasburnside]

Well, I couldn't solve it but at least I understood and appreciated the elegant simplicity of the solution.

[@anteeklund4159]

I actually managed to solve this one, and it was surprisingly easy

[@davidpham-e]

Thank you Mind your Decisions, this is actually one of the problems in my practice tests!

[@madhukushwaha4578]

If you want this type of Amazing questions that I will highly recommend this channel #mathsmining .

[@sharpmind2869]

3+π , I did it by just looking at the thumbnail and by the same method as yours. Very interesting problem ☺️👍👍

[@dave-in-nj9393]

the radius of the band is equal the the circumference of one circle. the 3 straight sides are each equal the the diameter of one circle. so total length is = 3 x diameter plus 1 x circumference

[@suver1able]

Finally one that I ‘saw’, 3 times 1/3 of the cirkel plus 3 times 1.

[@dro5819]

Yes thats what i thought too, then this guy comes up with 90°, 90°, 60° so this is 120°, making a simple thing much more complicated, thats why I hate maths

[@ManiKandan-dp1bg]

But 3(120)=360. ie)2×180=2×π. Then 1+1+1+2π?.first I understood clearly.Then confused about π=180°, and120+120+120=360=2π.pls anybody. Clear.

[@PYRAMIDHEAD1051]

Solved it in my mind elegantly.

[@karangupta1825]

Namaste, Presh. My name is Karan Gupta, I am from Ranchi, India. Could you please try solving this problem and if possible, then make a video: If 2^x=3^y=6^z, then what is the value of: 1)1/x + 1/y 2)x, y and z respectively. I used logarithms and my answers were: 1)1/x + 1/y = 1/{Log(2)Log(3)} 2)x = Log(6)Log(3) y = Log(6)Log(2) z = Log(2)Log(3) I also found that: 1/x + 1/y = 1/z.

[@araptuga]

New personal record: solved it before he finished showing/asking the question. (not expecting THAT to ever happen again!)

[@manusarda]

I guessed 6 by looking at figure. Proud to be that close 😀

[@tiltil9442]

You shouldn't be.

[@niebuhr6197]

Proud to be very wrong, ok dude

[@A909-g7g]

@@niebuhr6197 its actually not that far from og answer

[@mulbaybetul5924]

this is a very classic type of geo question about circle in our curriculum(Turkey). And there are more complicated versions of this question. If circles are touching (?) each other, it doesn't matter how many of them you have. Sum of the part of the circle areas always makes a whole circle. so after calculating this 1:21, you can just add an area of whole circle. But if there is blank between circles, then things change, you gotta calculate the angles and it doesn't always make a whole circle in this case.

[@ultrozy]

Love when he says "and *THAT'S* the answer"

[@AmericasAlgebraTeacher]

Good high school level problem- thanks! Please make more of these as they are more accessible to a larger audience of young mathematicians/geometers.

[@Advocatekamalkumarkarmkar]

His maths teacher is proud of him And Science teacher crying in the corner,🤣🤣🤣🤣🤣

[@ASAlan-sh8kf]

😂😂😂

[@KatlegoMasego]

Just woke up minutes ago and what a great way to start the day by learning new solutions.

[@matthiasburger2315]

The length of an elastic band? A very imprecise way of asking a question...

[@timothyball3144]

Right. It depends on how much stretch there is in the elastic band and how tight it fits.

[@kdmq]

Another way to work this out, is to realize that all curving points on the track are on circles at some location. Since you have to travel through 2*pi*n angle to end up pointing the direction you started, and since we clearly only turn around once when taking one circuit of the track, we have to have travelled through an angle of 2*pi. Therefore our total angular distance is pi since the radii of all 3 circles is 1/2. All you need to do is find that the 3 straightaways have length of 2 radii, meaning they have length 1. Therefore the total straight distance is 3. Our answer will just be 3+pi or 6.14.

[@lexus_bkl]

Presh: See you in next episode of MYD in where we solve the world's problems *one video at a time* . Me: Yeah, but how can someone solve two videos at *a* time

[@peterhawes9680]

It's pleasing that the curved parts of the band are just a little longer than the straight parts.

[@jimmykitty]

*The mathematical problem seems harder..... But actually its not correct at all* 😊 *I could solve the problem* ♥️🌿😜 *Love from Bangladesh* 🇧🇩🇧🇩♥️

[@jakkima1067]

Это самый лучший канал про математику! The best of the best!

[@part_time_photographer_]

Direct formula:. πd +nd n= number of circles d= diameter

[@karansandhu4827]

Ye jee waala hai kyaa

[@sandips1066]

Why does it not work for n=1

[@kitemg]

Solved it myself in under 10 sec with the same idea :-) That was the last light in my head today, now i go to bed :D

[@phoenixshade3]

2πr + 6r, just from the thumbnail, assuming the circles are congruent.

[@Grizzly01]

But why would you stop there, when the thumbnail gives a numerical value for ⌀, and therefore r?

[@phoenixshade3]

@@Grizzly01 1) It was the thumbnail on a cell phone notification, so it is literally smaller than 2cm. 2) General solutions are better than specific ones anyway.

[@Grizzly01]

@@phoenixshade3 1) Fair enough, you can have that. 2) if the question asks for a solution to a specific set of criteria, than that is what you answer. You wouldn't get full marks if this were a question in an exam, and you stopped short of providing the correct answer.

[@rajatpathak6524]

@@phoenixshade3 awesome man.....did you do it in head or used pen paper?

[@phoenixshade3]

@@rajatpathak6524 In my head; the band is encircling exactly 1/3 of each circumference and the tangential segments between circles each have a length equal to 2r, as they are parallel to the line connecting the circle centers and perpendicular to the radius at the tangent point.

[@smchoi9948]

If it's noted that the (unit) tangent vector along the closed curve (id. by the band) keeps turning (say clockwise) along parts where the curve overlaps w/ any of the circles, but maintains the same direction along complementary (i.e. "straight") parts, it's known that total length of the band's curved portions is exactly 1 whole circle's (clear to users of Gauss-Bonnet Thm.).

[@dashingd6163]

Am I the only one who finds this difficult ??

[@RudeGuyGames]

You'd think so, what with the astounding levels of insipid arrogance emanating from the comments sections of these videos, but I'm inclined to believe that this is not the case, although it's certainly an "easier" problem than most on this channel.

[@dashingd6163]

@@RudeGuyGames How did you do it exactly???

[@RudeGuyGames]

@@dashingd6163 I noticed that the parts of the string that follow the arc of each circle sum up to one circle's circumference and that each line tangent to a pair of circles was parallel to and of equal length to the line connecting their centers. It's hard to explain in text. Making a diagram and marking angles and relevant lengths helps.

[@dashingd6163]

@@RudeGuyGames Ok but you can never assume anything in mathematics ......... you clearly assumed that the sum of the string touching the circles aad up to 1 circle

[@JulianPlaysPiano]

A nice problem, but even easier than the explanation given! There's no need to identify the equilateral triangle and calculate the 60° or 120°. The parallel short sides of each rectangle show that each arc starts at the same angle as the previous one ends, so the arcs join to make a circle - regardless of how many arcs there are. So the combined length of the arcs is 2𝜋𝘳, in this case 𝜋. Then there is one straight line of length 1 for each circle, totalling 𝘯, where 𝘯 is the number of circles forming the ring. So the total length of the band is 𝜋+𝘯. And that's it!

[@jrkorman]

I'm going to stop watching now - Solved this one in exactly the same manner as Presh! Quit while you're ahead! 😃😃

[@callumross6290]

never quit learning! that’s how you start to forget the things you learned

[@jrkorman]

@@callumross6290 No kidding - I retired as a computer programmer just over 4 years ago. Sat down the other day to start coding up a new project and had to grab my books for a refresher!! At 66 I'm now understanding the old "I've forgotten more...." phrase.

[@blueneutrinostar]

(2pi + 6)r = 3+pi after watching the video and knowing that radius, r=half. Satisfying to solve it by looking at the thumbnail 😆 used the same method drawing the equilateral triangle in the middle followed by the rectangles. The only thing I did differently was I didn’t deduce the band was contacting one third of each circle by calculating the angles. I just intuitively thought that if it were 2 circles, the band would contact half of each circle, 3 circles -> one third, 4 circles -> one quarter, etc.

[@theb4stguy332]

Le engineers: No the answer can be further simplified to 6..

[@david-stephenmyles9539]

Now find the spring constants assuming the three circles are actually a 2D top view of three spherical stress/squeezy balls that are 70% compressed which perfectly balances the rubber band that is 20% expanded.

[@narada6329]

Lmao I actually managed to work that out using the thumbnail alone. Not that hard of a question, but still funny how quick I got it

[@adamae.7246]

Me too, I got pi+3 just by looking at the thumbnail and figured it out in my head !

[@joda7697]

I literally looked so fast that I didn't even see the diameter was given, assumed radius R for the circles, worked it out and then put R=1/2 once I did look. LOL

[@benbooth2783]

pi + 3 We can use this result to generate a general formula for the perimeter of the most compact shape with the smallest area that contains n circles. the circumference of the circle + (2 x the radius of the circle x n) A = 2 pi r + 2 n r A = 2r (pi + n)

[@suvanmadihally8914]

It depends how stretched it is

[@donvillamayor9133]

More relevant and practical solutions......younger ones will have an interest in mathematics when they see where it's use for.thank you.

[@reldahr01]

Next calculate the length of the band when it is unstretched

[@ZanyYooper]

A problem i could completely do in my head, nice!

[@madhukushwaha4578]

If you want this type of Amazing questions that I will highly recommend this channel #mathsmining .

[@soumyadip3751]

This was easy though 🙂

[@ridwansetiadi8393]

Try these and you'll find a pattern: 1. Draw 2 circles with the same diameter which touches each other, then calculate the length of the band around those 2 circles ! 2. Draw the same 2 circles again which touches the previous 2 circles (they form square-ish), then calculate the length of the band around those 4 circles ! If my calculation is correct the length of band (L) needed for n-amount of circle(s) with the same diameter (D) which is assembled to "kinda" form a regular n-gon (polygon with n-side) is: L = n*D + n*circumference/n L = n*D + circumference L = n*D + pi*D L = D*(n+pi) Which means (if I'm correct) no matter how much circles you add, the sum of the curved band will always equal to the circumference of 1 circle. Feel free to correct me if I'm wrong :)

[@kytexgd1463]

If it’s an elastic band, the length can change…

[@242math]

very creative problem, the solution is tricky and not complicated, great job

[@elementalic1520]

Yes, finally a problem I was able to solve!

[@madhukushwaha4578]

If you want this type of Amazing questions that I will highly recommend this channel #mathsmining .

[@nirajkumarverma5299]

Trick was in finding the angle subtended by the arc portion of band at the centre of circle. Explained very elegantly.

[@ps.2]

Nah. You know it adds up to 360° because it's a closed shape. No need to know the individual arc angles.

[@tanhakhandoker7045]

I kinda solved it with intuition. I'm glad that I got it right!!!

[@ahlpym]

pi + 3 The parts wrapped tightly around each circle cover 1/3 of each circle, so they combine to 1 circumference of the circle. The diameter is 1 so the circumference is pi. The remaining parts each go from the middle of one circle to the middle of the next (parallel to that, at least). So the length of each such segment is 2*1/2*d = d = 1. There are 3 of these, so these segments have a total length of 3. Put it all together, and you get pi + 3.

[@shufflex3360]

feeling proud for solving it after 4hrs 17mins

[@waheisel]

Thanks for the nice Father's Day present; an easy puzzle!

[@sciencegeeks1370]

I got it finally. Looks challenging but pretty easy . Nice

[@muhchung]

Finally I can solve something on this channel.

[@Dracopol]

Soooo easy. pi plus 3. You break it down into curves and straight lines and see the bits of rubber band are in contact with 1/3 of each circle, so add them up and they make a full circle, pi. Then 3 segments which are equal to the distance between centres of circles, or 1 each.

[@aabrightlove]

I did this a little differently: I imagined extending the lines touching the circles past their points of tangency to 3 common points, forming a triangle. From there it was some simple logic to deduce that this triangle must be equilateral, and therefore the minor arc intercepted by each of the three angles must be 120 degrees!

[@Rickety3263]

Woot! Solved it in my head in under 2 mins.

[@simonetozzi7912]

You don't need to calculate angles. The straight parts are 2r long each. The arcs are really one the continuation of the previous one (e.g. going clockwise) since the straight part is tangent to both circles, thus "maintaining" the same angle. Thus the 3 arcs summed up are a circumference: 2 pi r. Summing all together you get 2 pi r + 6 r. If now 2 r = 1 as the problem states, we get a total length of pi + 3. 🙂

[@jesusthroughmary]

This is so elegant.

[@randomsandwichian]

The circles are all with a similar diameter, and the length between each two middle points is equal to that, so 3 × 1 = 3 (diameter) The arrangement suggests that each point tangent to that middle points between the two poles is a third, ⅓ of a circumference. 3 times that arc equals 1 circumference. So the total length is 3 + π Edited my answer slightly since the calculations for a circumference of 0.5 radian is 3.14, ie. π.

[@vedants.vispute77]

No matter how many circles form a closed loop, their curved surfaces add up to a circle!

[@akhil6826]

Really? Can u please attach the link to the proof of this

[@vedants.vispute77]

@@akhil6826 take ∞ circles, their curved surfaces would be ∞ points and those ∞ points will join ∞ times and will make a circle.

[@benjamininkorea7016]

Wow I got this one right away! Nice to feel smart for once.