The Three Square Geometry Problem - Numberphile

[@strongeststrike6737]

‘This problem has not only 1, not only 2, but at least 54 solutions.’ That escalated quickly

[@nessfrommother2147]

Pythagorean Theorem: am I a joke to you?

[@GamerShen98]

This is extremely easy question if all sides are equal u can put X as length .. then in the beta angle just do Pythagoras as the base is 2X and the hight is x , then the hypotenuse =sqr of 5 and then shift sin/cos/tan give you the angle and u do the same for the third one and thats it

[@ZachAttack6089]

@@GamerShen98 Sine and cosine are irrational, so they don't give exact answers. So you'll end up with something like 90.000000003, which doesn't "prove" that it's 90. Also, this is geometry so I don't even know if you're allowed to use sine and cosine.

[@kamil.g.m]

@@ZachAttack6089 So are you saying tan(45) is not 1, but it's actually 1.0000000003?

[@ZachAttack6089]

@@kamil.g.m Not tan(45), but most values of sine and cosine and tangent are inexact, so they can't be used to "prove" anything. Besides, it's geometry not trigonometry. I don't think you're even allowed to use trigonometric functions.

[@jk844100]

Wow, this whole thing felt like the climax of a Phoenix Wright case except instead of finding a murderer you're finding an angle.

[@vincent-of-the-bog]

A CONJECTURE!

[@mads_in_zero]

[Mia Fey Voice] Phoenix, turn your thinking around! Don't ask what the angle has to be. Just assume what a 5th grader would know!

[@uuu12343]

Eq_NightGlider_ CONJECTURE! *slams table* I beg to disagree, Angle-o

[@cptaincrunch4438]

Angles are sharp

[@Lengo67]

It was Colonel Mustard in the dining room with the right triangle.

[@nameguy101]

"The size does not matter" - A renowned professional scientist Take that, society

[@LegendaryFartMaster]

This maybe be six years too late, but this comment is underrated.

[@rockifythis]

Thanks, that makes me feel better about myself

[@machineman8920]

don't say the s word

[@samisiddiqi5411]

Okay, Coomer.

[@migfed]

I like her style, she explains the problem as telling you a detective story, it thrills you! Greetings from Colombia.

[@ivpantev]

This video is brilliant not because of the problem, but because it shows you the thinking behind reaching a solution. Everyone who said to use trigonometry is not wrong per se, but simply taking a more complicated route to reach the same conclusion. The beauty here is that you can get where you want to be just by drawing a few lines rather than using advanced functions, i.e. you can solve a much more complicated problem with simpler tools and some creativity :)

[@nsq2487]

In the real world where I have a calculator and know about inverse trigonometry, I would use the so called "complex" method any day. Yes, the constructing lines solution was elegant, but not practical. Trigonometry was meant for problems like this and it would be the first thing to pop into the mind of any engineer. It is the most simple solution, find each of the angles, add em up , done. Just because it requires a calculator doesn't mean it's complex.

[@freddiehand6551]

@@nsq2487 In practical use, trigonometric methods are indeed the most efficient, and is thus used in engineering. However, this is only a maths problem intended for recreation, where we are not concerned about the speed of reaching the solution.

[@freddiehand6551]

@@nsq2487 in addition, you would have to use trigonometric identities to verify that the angles add to exactly 90 if you are only in possession of a pocket calculator, because it is possible that the sum is very close to 90 so that the calculator round the answer, but it is not exactly 90.

[@heronimousbrapson863]

I Pantev So much easier to use trigonometry. 5th graders are not normally taught trigonometry though. Otherwise, you could measure with a protractor. Cutting pieces of paper and putting them together seems more like art class.

[@keescanalfp5143]

@@heronimousbrapson863, A nice thing is that children aged up to 14-16 love seeing you folding, cutting, adding real material to illustrate no matter what. Even geometry.

[@superj1e2z6]

I give this video 7 points.

[@zchelmerjoashgamboa7366]

nice reference.

[@NoriMori1992]

Ohohohoho!

[@composto]

Oh, nice one!

[@hblanco530]

Is that an imo reference?

[@dominikskorjanc]

Nicee😂

[@yugyfoog]

This problem has a very simple solution using complex numbers. (1+i)(2+i)(3+i) = 10i, which has an argument of 90 degrees. Of course I didn't know about complex numbers in fifth grade.

[@justpaulo]

Your solution is the most elegant, non-geometric, solution of all ! True that you don't know complex numbers in fifth grade, but you don't know trig either :) In fact it makes me think how good it would have been if, when I learned complex numbers, such a practical example was given to me.

[@foobargorch]

Beautiful!

[@timh.6872]

yugyfoog I feel bad for necrobumping, but that is a beautiful thing, and yet another reason why we should introduce complex numbers right after fractions alongside the irrationals. Not only does it make algebra easier (being an algebraically closed set), it's really stinking useful for plane geometry.

[@banjonpro5649]

can someone please expand on this solution?

[@dragonite7780]

Complex numbers have the property that if you multiply 2 complex numbers, their arguments add together and their modulus' multiply together. yugyfoog exploited the adding angles part.

[@keithwilson6060]

The best part of this video was her accent.

[@joshlewis8753]

@Paco Bulgarian

[@mariostar13]

@Paco Yep! Definitely Russian!

[@brenmcclure128]

Yes Dr. Jones

[@gmortimer20031]

@masonery123 Just for a fleeting moment, but one which extended to a point at infinity in a hidden dimension, I wished I was a single man!

[@ericastonehouse9979]

definitely lol

[@gabcastel]

i love absolutely love her accent.

[@tomasvolejnik2809]

i hate it

[@FLomasterZ]

It's Bulgarian accent and it's very similar to Russian. ;)

[@Martial-Mat]

Jeez - if that's her idea of an easy solution that a 5th grader should be able to work out, she went to a different school to me! I don't think we touched algebra till 8th or 9th grade and cancelling of elements till a year or two after that. It was elegant though, and she's very pleasant to listen to.

[@whychoooseausername4763]

A lot of mathematically gifted kids were pushed into competitions for maths or physics. I did competitive maths physics and literature just for fun throughout middle school and high school. We were often asked difficult questions and asked to produce original proof.

[@travisbaskerfield]

Mat Broomfield. The deliberate dumbing down in American education?

[@mequable]

Bulgarian education was crazy back then. Now kids are lucky to be able to write properly when finishing 7th grade.

[@Martial-Mat]

Oh really? Why do you think that is pastichka?

[@GRBtutorials]

Well, here in Spain, we are introduced to algebra in the equivalent to 6th grade, and I personally knew a bit of it by the equivalent of 5th grade as well.

[@FlyingTurtleLP]

I really love vids like that. Will never regret that I subscribed this channel.

[@numberphile]

FlyingTurtle thanks - we're happy to have you as a subscriber

[@zacharygeorge4595]

Numberphile I love this channel! I'm a number nerd, so I learn everything about numbers as possible. I've watched like every single video!

[@aghaanantyab]

actually this channel is dedicated for psychopath. if you like this channel it means that you could potentially be a psychopath

[@aghaanantyab]

Thomas Anderson you're welcome :)

[@fishyeverything8530]

Me too 😄

[@numberphile]

There's a lot of links to go with this video (extra footage, associated video, brown paper, discuss on reddit, etc)... See the full video description for all these links.

[@randomguyontheinternet801]

.An underrated comment>

[@headrockbeats]

What scares me is that we only learned this sort of stuff in 7th grade, in geometry class for _advanced_ students...

[@HoratioAccel]

+Headrock In the US Geometry is usually taken by Sophomores in _high school_. It's a sad world.

[@water7648]

+TheCycloneRanger im in 7th grade doin geometry but im da only 7th grader for it which is really sadddddddddddddd....

[@HoratioAccel]

Turtle cat For high school credit? If so, damn, I only knew a handful of people at my middle school who did that.

[@water7648]

not for high school credit. IM in middle school but they let me get into geometry since I passed algebra last year in six grade

[@HoratioAccel]

Turtle cat If it isn't for high school credit (and shows as such on your end of year transcript) then you are more than likely going to have to take it again in high school. I would check this if I were you, just in case.

[@iammaxhailme]

arctan(1/1) + arctan (1/2) + arctan (1/3) = 90 A bit easier, but less interesting than the video's method, of course.

[@usurper1091]

yeah

[@ptyamin6976]

you have to refer to these trancendental functipns which are not as elementary, and thus elegant, as simple geometry

[@stevenvanhulle7242]

I guess you used your calculator for this. That's not a way mathematicians do it. (Especially in inverse geometric functions) calculators make small errors, and you may end up with a result of 90.00 degrees, while in reality it could be 89.9999999943. In that case the conjecture doesn't hold. For instance, it's impossible to construct a regular nonagon with only compass and unmarked ruler. Yet there is a construction which comes real close, something like 99.999999999999%. Your calculator may incorrectly suggest that it's a valid construction for a perfect regular nonagon, but like I said, it's impossible.

[@shvoregavim9435]

Steven Van Hulle Just use the formula for sum of arctangents.

[@meJevin]

yeah, without a calculator you wouldn't be able to calculate this, so you would have to use an arctanget summation formula. you will get something like this if you do: arctan(1) + arctan(1/2) + arctan(1/3) = arctan(1) + arctan((1/2 + 1/3)/(1 - 1/2 * 1/3)) = arctan(1) + arctan((5/6)/(5/6)) = arctan(1) + arctan(1) = pi/4 + pi/4 = pi/2; (or 90 degrees if you prefer)

[@ItsClint]

This is brilliant. Loved both this and 'Pebbling a chessboard'. I'd love to see more videos from her. Keep up the good work, Numberphile!

[@lammatt]

wow... this is brilliant.

[@orbemsolis]

rilesthegiles I have completely forgotten what we were discussing

[@hellje]

matt lam жцжцжц

[@eldiospadre128]

Aw man, I haven't worked with geometry in years! This video took me back! Thanks for posting this it!

[@numberphile]

Diego Rivera you're welcome

[@Maverikmkd]

I was always amazed of all mathematical problems where you have to think "out of the box" in order to solve the problem. That is the most difficult skill to get when you participate math contests, and I was also amazed by all kids who had those kind of skills. For me, it was just magical and astonishing. I am still wondering, how to train kids to think like that?

[@lhopitalified]

There's also an elegant solution via complex numbers: Let A = 1+i, B = 2+i, C = 3+i. Then the argument (angle) of the product ABC is the desired sum alpha + beta + gamma. Since ABC = 10i, the sum of the angles is pi/2.

[@aminsust1236]

ru

[@tri5ford]

Approx 3:10: "Or some ugly angle nearby?" Priceless...!

[@Sirenhound]

I paused it until I figured it out. My solution was a little different (one of the other 54 I suppose): I chose the top left corner where all the lines radiate from; The 45 and the 18ish are already there, so to solve I need to show that the remainder is the 26ish. I mentally extended the diagonal (45) line down to be twice as long (terminating directly below C), and bring it up from that point to corner D. this constructs a similar triangle at sqrt(2) scale, seating the 26ish angle neatly in the desired position. Q.E.D.

[@DewZJ1997]

"The size does not matter" -Zvezdelina Stankova2014

[@manukrishnanms1317]

Time stamp please

[@xaviconde]

"Exaaaactly"!! 😂

[@lucasschuetz2111]

It started so simple, then grew to something so exciting. Love these videos! Easily my favorite channel, keep it up!

[@irwinwinaris9800]

Just brilliant that math can be so simple yet so fascinating.

[@qbwkp]

"The size does noy matter:"

[@valeriobertoncello1809]

I was just looking for a comment like that ahahah it didn't take much time

[@JamBos11]

Only the proportion matters 😉

[@randomaccessfemale]

If you are worried about size, just change the metric. That's what I always say.

[@NoriMori1992]

"noy"

[@f1f1s]

Two sample problems from a Soviet quiz for second-graders (that is, 8-year-old children): 1. A digital clock displays hours and minutes (e.g. 13:08). Little Tim loves the digit “5”, and he enjoys watching it on the display. What is the longest possible continuous time span during which Tim can revel the sight of the number “5” on the display? 2. Theo picked two 3-digit numbers with equal sums of digits. He subtracted the smaller number from the larger one. What is the largest possible difference Theo could obtain? Now you get it why Soviet mathematicians emigrated to the USA and Europe after the fall of the Iron Curtain? Because their skills and wit were highly demanded in the West.

[@weekendresearcher]

The 1988 IMO Problem no 6 killer!

[@taweller]

Professor Zvezdelina Stankova + that blue dress = Nice combo!

[@allluckyseven]

Beautiful problem. I wonder to how much would the angle tend to, if you kept adding squares to the right.

[@girgakos]

you can also say that tanβ = 1/2 and tanγ = 1/3. Then you say that tan(β + γ) = (tanβ + tanγ)/(1 - tanβ*tanγ) => tan(β + γ) = 1 => tan(β + γ) = tan45° => β + γ = 45°. So α + β + γ = 90°

[@johngowers5403]

A method I found that doesn't require drawing any more lines on the diagram: Go to the diagram at 1:08 on the video. Look at the top left hand corner. It is divided into four angles by the three lines that meet that corner. The leftmost of these four angles is clearly equal to alpha (isosceles triangle). The topmost is equal to gamma ('Z' rule). Therefore, it suffices to show that the two angles in the middle sum to beta. For this, we spot a pair of similar triangles in the diagram. If we label the vertices along the top A,B,C,D and the vertices along the bottom W,X,Y,Z then we claim that the triangles AXY and AXZ are similar. Indeed, they share the angle

[@FirstLast-dd8ff]

This video was amazing; really liked the simple animations and the woman's voice felt nice to hear. Awesome!

[@numberphile]

First Last that is nice of you to say

[@EugenAugustinus]

It's really a very beautiful problem. I think the most simple proof can be given with complex numbers. You can write (1+i)(2+i)(3+i) = 10i (i^2 = -1).

[@MrV1604]

This is so awesome I cried .. I'm showing this to my little sister !

[@luksiv]

every single video of Numberphile leaves my mind blown of how someone can come up with these kind of ways of solving such complex problems... kudos for that !

[@kipshiux333]

Professor Stankova is SO engaging, I really enjoy videos with her. Brady, can we have more of her, please?

[@canalf007]

slopes of the 3 lines are tan(a)=1=45°, tan(b)=1/2, tan(c)=1/3. Let x = a+b+c. Then x-45° = b+c. Applyint tangent both sides: tan(x-45°) = tan(b+c) tan(x-45°) = [tan(b) + tan(c)] / [1-tan(b)tan(c)] tan(x-45°) = [1/2 + 1/3] / [1-(1/2)(1/3)] tan(x-45°) = 1 x-45° = 45° x = 90°. ggwp

[@Kourindouinc]

As Zvezdelina Stankova stated, there's several ways to solve this problem, and several of them uses trigonometry. Another alternate solution, although it is similar to the video, has an alternate way of proving that EHD is a right angle. Let's assume the side of a square is of x. The slope of EH is x/2x or 1/2. The slope of HD is -2x/x or -2. If two slopes are opposite to each other, meaning one is x and the other is -1/x _(Where x here would be 1/2),_ the two segments are perpendicular, meaning EHD is exactly 90 degrees. Proving this theorem is as easy as the step they used instead. There's a lot of shortcuts that one can do while doing this, although a fifth grader may not have access to all of them.

[@michaelbauers8800]

Where does someone get those mega tools she's using?

[@qorilla]

You can also collect them in the top left corner. The leftmost angle is 45°, the middle two add up to beta and the rightmost/topmost angle is gamma. Gamma because it's just a flipped version of gamma triangle up there, and beta because if you extend the lines downwards you can make a 2 by 1 triangle diagonally. Too bad I can't post a pic.

[@Seth4All]

Brilliant solution. Very elegant.

[@numberphile]

TheMathKid glad you liked it

[@DjVortex-w]

That was an amazingly complicated way of explaining the six-square solution.

[@Michael_Arnold]

And so, so slow!

[@Michael_Arnold]

And so, so slow!

[@mullergyula4174]

I guess they wanted to make it easy for all audiences.

[@willdeary630]

0:19 "Size does not matter", you heard it here folks.

[@adamplace1414]

I saw Adam Savage referred to this video as one that moved him emotionally, and I can see why. This was great.

[@Xalnop]

Great great great great video! I love it.

[@numberphile]

Xalnop thank you

Let me just point out that I loved this professor. Her demeanor, her choice of words, the way she explained conjecture and of course her accent really makes it fun to listen to.

[@SomeDude881]

That method was thinking right out of the box literally. This video impressed me for such an unique way to solve problems.

[@jujumas8708]

This is the math people should learn at school, not the one that is filled with doing the same things over and over again.

[@dr.mikelitoris]

Juju Mas it is what we learn at school though

[@AllieAndPeach]

some people need repitition to really learn and remember a concept. nothing wrong with that.

[@BonziFedoraINC]

Sam Harper ah someone that gets it

[@DeathlyTired]

I remember Prof. Stankova from the 'Pebbling a Chess Board' film. Again, she really is excellent; clear, concise & thorough.

[@ButzPunk]

Her accent sounds so familiar to me... is she Bulgarian? (I might be completely wrong)

[@nandvinchhi1217]

ya bulgarian olympiad winner

[@johembrey3616]

The three square problem, and the six square solution. Beautiful.

[@bombasticbrian.]

Math is a beautiful subject

[@wafikiri_]

There is a much simpler proof. No need to draw three other squares and a slanted big triangle. Just get a diagonal DK in the third square (new labeled vertex K) CDJK. Then all the figure has a 180º rotational symmetry, and the square with vertices A, B and E has diagonal symmetries, of course. Then, at vertex E, we have: angle JED = angle EDA = gamma; angle DEC= angle ECA = beta; and angle BEA = angle EBA = alpha, the three of them composing angle AEJ = 90º. And the said diagonal DK has not even been used, except to show the 180º symmetry. Update: typo correction.

[@TheDiggster13]

Well suddenly, my solution involving the sine and cosine rules, and pythagoras' theorem seems horribly inelegant!

[@StEeEpPhen]

angle(AEB) = 45 d. ; The first square and the one next to it makes a rectangle, meaning AC parallel to EX (note: X is the point of the third dot, right above C). AC || EX meaning beta=angle(ECX)=angle(CEX) because those angles are alternate interns angles. Now we have the ADJE rectangle. AD || EJ meaning that gamma = angle(ADE) = angle(DEJ), because those angles are alternate interns. If we sum them up: alpha + beta + gamma = angle(AEB) + angle(CEX) + angle(DEJ), but gamma = angle(DEJ) = angle (BEC) (Note: further demonstration is needed for this affirmation, but I won't do it; it's pretty easy to demonstrate). Now, the equation becomes: alpha + beta + gamma = angle(AEB) + angle(CEX) + angle(BEC). But once again, angle(AEB) + angle(CEX) + angle(BEC) = angle (AEJ). The angle(AEJ) = 90 d. (square angle), so the sum: alpha + beta + gamma = 90.

[@chadtindale2095]

Proving space using Negative Space. I've never seen that applied to angles, but it's blowing my mind. And I love it.

[@naor5683]

WOW we should all be thankful to the greeks for trigonometry xD

[@karrieadler2808]

I actually came up with a somewhat simpler solution, It's very close to this though. I start with the single square and draw the line to the corner to create that angle, I draw 2 more squares above it (3 high in total) and draw a line from that point down to the same corner. then if you imagine the original 45 degree line as a side to a square and draw the rest of the square, and then draw another square towards the point at the top of the stack of smaller squares, it lines up with point at the top (showing that the middle segment is 2 squares apart) and all the angles fit into the 90 degree corner yay, (It's a lot simpler to show visually than to explain)

[@archimedesworld3202]

@jimpikles She said it was 5th grade. Here is why you DON'T jump to trig with sin and cosine etc... Its bad practice.. why bother explaining arithmetic to kids.. just teach them long division and forget about the reason it works or why. See the point? It's about understanding not about passing a test fast. Understanding is far superior to memorizing formulas. Creativity will get you further. The age of genius Einstein's and Euler is gone precisely because understanding has been thrown to the side for the sake of practical or commercial use.

[@kevinworner4083]

ArchimedesWorld The imagination is being reduced to almost non existence

[@angelmendez-rivera351]

Well, historically, schools were never meant to teach. The governments are perfectly fine with not having children learn as it would be inconvenient for them to have children learn only to overthrow them later. Private schools are not much better for the exact same reason. It is unfortunate.

[@trungkiennguyen9193]

I actually found another way to solve it using similar triangle. If i name the point of the original shape ad A, B, C, D, E,F,G,H with the 3 original line being AG, AF and AE then triangle AGF is similar to ACE because AG/AC=GF/CE=AF/AE. This can be prove quite easily if you set the side of the square as x and then calculate using the pythagorean theorem

[@trungkiennguyen9193]

It surprised me not to see anyone in the comments with sth else other than trig:))))

[@keescanalfp5143]

@@trungkiennguyen9193, Thanks a lot! We see that John Gowers did almost the same way now '3 years ago'. Can't see the exact date. Two similar triangles with ratio √2, all within the original three squares and the original drawn lines. In your names, triangles AGF and EGA similar with their side ratios √2 : 1 resp. 2 : √2 and common angle G. Then angle EAG = bèta, the 26ish, and the three appear in a row at A. Note that Mrs. Numberphile later on uses an other lettering.

[@akanegally]

Another way to demonstrate it arctan(1)+arctan(1/2)+arctan(1/3) = 90°

[@PeterAuto1]

but therefore you need the exact values of arctan(1/2) and arctan(1/3)

[@Shadowmere29]

+Peter Auto no you don't need the exact values. Just use the tangent angle sum formula

[@reetasingh1679]

+Peter Auto You can use the tan(x+y) identity... then you will get (1/2+1/3)/(1-1/6) which is equal to 1, therefore sum of the two angles is 45

[@sanyamahuja7896]

only if fifth graders knew trigonometry

[@Shadowmere29]

Lol

[@Jodabomb24]

Professor Stankova might be my favorite professor featured on this channel. She always presents the information clearly and in an interesting way; you can always see how much she loves what she's doing, and that's really important. I just love listening to her. And yes, her accent is fantastic :D

[@robinbuster1323]

hypotenuses is correct. Singular: υποτείνουσα Plural: υποτείνουσες

[@RomeoUW]

This is pretty awesome and really understandable due to the way it's explained. Thanks for sharing with us this beautiful knowledge !

[@Chriib]

45 degrees + 2/3 of 45 degrees + 1/3 of 45 degrees. That was my first thought. It is not the right approach but the answer is correct.

[@christianrodriguez823]

These are some of the most brilliant solutions, the ones that are rooted in basic geometry and algebra but require such outside-the-box thinking. As an aspiring teacher, I believe these are the kinds of questions we should be giving to our children in school and I hope to give my future students some problems like these to open their minds and challenge them to think differently about math.

[@raoulhery]

Can I use Trigonometrics here?

[@Fun_maths]

You get arctan(1)+arctan(2)+arctan(3) Not clear that it sums to 90 degrees

[@sudeshkumar2222]

it's really awesome ......lack of visualization I couldn't solve this first time...now I smoothly understand.....thnx

[@venkybabu8140]

Try Ptolemy. Try sin waves. Try gears. Try angle numbers. Euler's line. Try log. The area of the big is equal to sum of the other Selenes.

[@milosmitrovic1271]

8:35 Those two lines (EH, HD) were catheti or Cathetuses of that triangle, and the bottom line (ED) was a hypotenuse. One of my favorite solutions, very nice work. :)

[@BohonChina]

many ways to prove, another way I modified is the equal length of adjacent side of big triangle can be the square root of (1 unit squared plus 2 unit squared) is the square root of 5 the hypotenuse of triangle is (1 unit squared plus 3 unit squared ) is square root of 10 5 + 5 = 10. that angle must be 45 degree.

[@jamesleng8210]

Here's another solution I really like: The problem is equivalent to: arctan(1) + arctan(1/2) + arctan(1/3) This is equivalent to: arg(1 + i) + arg(2 + i) + arg(3 + i) (where i = sqrt(-1)) which is equivalent to arg((1 + i)(2 + i)(3 + i)) because multiplying complex numbers sums up their arguments. Multiplying, we have that: arg((1 + i)(2 + i)(3 + i)) = arg(10i) Since 10i is imaginary, the argument of 10i is pi/2 = 90.

[@rhythml6229]

Secondary school level okay... even this way of solution

[@spotted_stingray]

or you can just use foumula. sin(b)=1/sqrt(5), cos(b)=2/sqrt(5) sin(c)=1/sqrt(10), cos(c)=3/sqrt(10) sin(b+c)=sin(b)cos(c)+sin(c)cos(b) =1/sqrt(2) therefore b+c=45°

[@alexandterfst6532]

Interesting problem, here's the way I approched the problem. tan(Beta) = x/2x =1/2 tan(gamma) = x/3x = 1/3 if Beta+Gamma = 45° (pi/4), then tan(Beta+Gamma) = 1 but tan(Beta+Gamma) = (tan(beta)+tan(gamma)/(1-tan(beta)tan(gamma)) = (1/2+1/3)/(1-1/6) = 1 !

[@MrNemay]

So a fifth grader is supposed to know all that ?? :O

[@ZakX11]

Ne May Well , a 5th grader should know the simple geometry used in the video , but probably not how to manipulate it so much. It's like you have simple tools which you know how to use , but it takes skill to make something beautiful using them.

[@DeltaHedra]

What I find more amazing about this problem is that the initial red lines which join the corners of the rectangles, where they touch the uprights of the squares are the points of a parabola.

[@hj8607]

Simply turn the entire first three squares and three lines upside down superimpose on original diagram . (you now have 3 squares and 5 (five) diagonal lines ) IN THE right hand square you now have a diagonal at 45° and below it two lines that duplicate the original two lines (one is in same location as first and second goes up one square and over two squares, just as original ) and the sum of those angles (plus the 45 ) equals the square corner or 90 ° . Truly the most simple 'magic ' solution . The video explanation it the LONG way around .

[@cursedswordsman]

USING ARCTAN IS NOT A PROOF UNLESS YOU USE A FORMULA TO SHOW IT ANALYTICALLY. ARRRGH

[@stellarwinda]

cursedswordsman Indeed, it seems half of the people here think that taking arctan on their calculator is fine.

[@paulbin]

cursedswordsman You are wrong my Little friend

[@liberphilosophus7481]

Hue hue arctan :P

[@dominantwolf4593]

Just use regular SOH-CAH-TOA on each angle and add. Assuming you don't have to prove your unit circle rules true which isn't that hard either.

[@dominantwolf4593]

+Harry Potter granted I'm a chemist so I believe integral are like addition lol

[@kennethmccormick1791]

Thank U Numberphile. Brought back some good memories. Now, it's all Excel sheets :(

[@GMwoogitmaster18]

This solution to the conjecture was absolutely elegant...but I still tried to brute-force use trig to solve it, too.

[@calvindang7291]

so i guessed 82.5.

[@MrJuanideluxe]

Calvin Dang 45+45/2+45/3? :P

[@JohnSmith-ry3rp]

In my opinion, the most logical way to go about this involves basic trigonometry. If the squares are all equal in length, then you can get the answer by using the tan^-1 function. If you look at it, the inverse tangent ratios for the three angles are tan^-1(1/1), tan^-1(1/2) and tan^-1(1/3). If you add them up you get: tan^-1(1/1) + tan^-1(1/2) + tan^-1(1/3) = 90 Try it yourself it works.

[@TheWindWaker333]

Aside from adding three more squares to work with, which is thinking outside the box, I don't see how this problem is complicated. The geometric approach of moving around the triangles is certainly more beautiful and intuitive than just adding up the three arctans though.

[@shield543]

Awesome video :), Oh and first?

[@billzorbas3889]

Young girls fishing

[@AlexisGelinas]

I freaking love the geometry videos on numberphile.

[@AlexisGelinas]

And I do think a fifth grader could totally understand this, especially those that like puzzles.

[@joeyhardin5903]

i just used trigonometry because i noticed for the second square the line comes in exactly half way through the square and on the third one it comes in exactly one third in. With a calculator, type in 45 + atan(1/2) + atan(1/3) and you'll get 90.

[@nrxzionistlibertarian6168]

Simplest solution: arctan(1) + arctan(1/2) + arctan(1/3) = π / 2 = 90°

[@NotTechLead]

Because the average 5th grader knows what arctan means?

[@nrxzionistlibertarian6168]

Well, the average 5th grader doesn't get such tasks either.

[@dyld921]

libertarianDE This is a 5th grade problem, as mentioned by the professor

[@sSunbeamM]

Dylan Dang everybody STOP SCREAMING!!!!!!!!!!!!

[@sonihi4]

Well an average 5th grader could understand the functions

[@macieyid]

This is the key to nice formula for approximating π using arctg series: arctg ½ + arctg ⅓ = arctg 1 = π/4

[@chrisnachos22]

0:19 if only all women thought this, am i right men? 2:41 if only we could all make women make this noise, am i right men?! 3:24 mouse in the house

[@davidmoseley2277]

OR.... Assuming 90 degrees for each segment 90 - A (45)=45. 90 - B (26)=64. 90 - C (18)=72. Therein lies the difference of 45 + 64 + 72 = 181. Add 181 to 89 gives you 270, or 3x 90 = 270. Works for me.

[@giordy9013]

The wrong beta moment was the funniest moment of almost all numberphile history

[@XTownCuber]

rubik's cube ftw

[@Pingstery]

Came up with 2 solutions just after the problem was explained, this indeed is a brilliant problem that allows kids to apply what they've learned in their own ways, instead of forcing a teacher's decided approach. I wish this was shown to me in my geometry class! Definitely going to forward it to my old teacher, hehe.

[@ericvega9160]

am often intrigued at how people can remember very complex procedural problems yet they tend to forget very simple stuff .

[@sl4yd]

I wish I were HIGH on POTenuse

[@681726]

Thanks for the problem. I worked out the solution after the hint @5:19-5:25.

[@powong4136]

The subsequent 4th angle can also be obtained as 14.03624347...degrees The 5th angle =11.30993247......degrees The 6th angle = 9.462322208......degrees The 7th angle = 8,130102354......degrees The 8th angle = 7.125016349......degrees The 9th angle = 6.340191746.....degrees The 10th angle= 5.710593137.....degrees The 11th angle= 5.194428908.....degrees The 12th angle= 4.763641691.....degrees The 13th angle= 4.398705355.....degrees The 14th angle= 4.08561678.......degrees The 15th angle= 3.814074834.....degrees The list can be continued up to the n th angle = ArcTan(1/n) And when n=Infinite, the angle = 0 Please note that the main objective of the lecture is to teach the sixth graders( or lower graders) how to use an angle protractor to measure angles and sum up all those angles with prediction of the outcome into the unknown, Therefore, this is a excellent lecture for other educators to learn how to teach other students who have no profound knowledge of mathematics.

[@JonMcAwesomeness]

arctan (1)+arctan (1/2)+arctan (1/3)=90° ♡

[@CrazySteve113]

+Johnny Except you can't calculate those with simple arithmetic.

[@AustinGarrett777]

Conveniently, the slope of each line segment is given, which allows you to find the relative sizes of each side, and calculate the angles through trigonometry.