The Three Square Geometry Problem - Numberphile
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- čas přidán 17. 09. 2014
- Three Square Geometry Problem
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Featuring Professor Zvezdelina Stankova.
Extra footage1: • Squares & Triangles (E...
Extra footage 2: • Squares & Triangles (E...
Pebbling a chess board: • Pebbling a Chessboard ...
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‘This problem has not only 1, not only 2, but at least 54 solutions.’
That escalated quickly
Pythagorean Theorem: am I a joke to you?
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
@@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.
@@ZachAttack6089 So are you saying tan(45) is not 1, but it's actually 1.0000000003?
@@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.
Wow, this whole thing felt like the climax of a Phoenix Wright case except instead of finding a murderer you're finding an angle.
A CONJECTURE!
[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!
Eq_NightGlider_
CONJECTURE!
*slams table*
I beg to disagree, Angle-o
Angles are sharp
It was Colonel Mustard in the dining room with the right triangle.
"The size does not matter" - A renowned professional scientist
Take that, society
This maybe be six years too late, but this comment is underrated.
Thanks, that makes me feel better about myself
don't say the s word
Okay, Coomer.
I like her style, she explains the problem as telling you a detective story, it thrills you! Greetings from Colombia.
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 :)
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.
@@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.
@@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.
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.
@@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.
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.
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.
Beautiful!
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.
can someone please expand on this solution?
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.
I give this video 7 points.
nice reference.
Ohohohoho!
Oh, nice one!
Is that an imo reference?
Nicee😂
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.
.An underrated comment>
The best part of this video was her accent.
@Paco Bulgarian
@Paco Yep! Definitely Russian!
Yes Dr. Jones
@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!
definitely lol
I really love vids like that. Will never regret that I subscribed this channel.
FlyingTurtle thanks - we're happy to have you as a subscriber
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!
actually this channel is dedicated for psychopath. if you like this channel it means that you could potentially be a psychopath
Thomas Anderson you're welcome :)
Me too 😄
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.
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.
Mat Broomfield. The deliberate dumbing down in American education?
Bulgarian education was crazy back then. Now kids are lucky to be able to write properly when finishing 7th grade.
Oh really? Why do you think that is pastichka?
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.
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!
i love absolutely love her accent.
i hate it
It's Bulgarian accent and it's very similar to Russian. ;)
She is so great at explaining! I think even people, who didn't work with angles for long, will understand it :)
It started so simple, then grew to something so exciting. Love these videos! Easily my favorite channel, keep it up!
arctan(1/1) + arctan (1/2) + arctan (1/3) = 90
A bit easier, but less interesting than the video's method, of course.
yeah
you have to refer to these trancendental functipns which are not as elementary, and thus elegant, as simple geometry
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.
Steven Van Hulle Just use the formula for sum of arctangents.
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)
What scares me is that we only learned this sort of stuff in 7th grade, in geometry class for _advanced_ students...
+Headrock In the US Geometry is usually taken by Sophomores in _high school_. It's a sad world.
+TheCycloneRanger im in 7th grade doin geometry but im da only 7th grader for it which is really sadddddddddddddd....
Turtle cat For high school credit? If so, damn, I only knew a handful of people at my middle school who did that.
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
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.
Aw man, I haven't worked with geometry in years! This video took me back! Thanks for posting this it!
Diego Rivera you're welcome
I am so glad to see anothervideo featuring Dr. Stankova. The problems she takes us through are always fascinating. Simple questions with complex implications.
I can't wait to see the next one!
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 !
"Oh, it's a little bit obtuse" - my teachers about me in 5th grade
Speaking as an architect who makes a living working with geometry, this is brilliant. Also just another reason to love the number 6.
More geometry videos!
really enjoyed this talk and i think the teacher is quite amazing too. Watched all three parts and the discussion was fun, light and interesting!
This is pretty awesome and really understandable due to the way it's explained. Thanks for sharing with us this beautiful knowledge !
I wanted to try to resolve the problem myself just for fun, so I paused the video and I spent like 10 min on it. I really like the fact that even though the basis of the method is the same, I still resolved it in another way than Pr. Stankova did. And it was so fun to do. Basically I multiplied the squares to a small grid, then I recreated a similar construction but with the diagonals of the first squares as the sides of the new squares. I had then new 90° angles and with the use of all the parrallel lines I could put together alpha, beta and gamma in one of the 90° corner and they fit perfectly. Proving then that their sum is 90°.
I love how you can manipulate and distribuate angles using parallel lines. It's like the energy of the 2D geometric world x)
I paused the video too to try and work it out for a minute or two. Came up with 90 as a guess
wow...
this is brilliant.
rilesthegiles I have completely forgotten what we were discussing
matt lam жцжцжц
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?
Just brilliant that math can be so simple yet so fascinating.
Solving this as a 5th grader and being one of 11 out of 260 kids to solve the infamous "Problem 6"..... much respect
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.
Professor Stankova is SO engaging, I really enjoy videos with her. Brady, can we have more of her, please?
That method was thinking right out of the box literally.
This video impressed me for such an unique way to solve problems.
This video was amazing; really liked the simple animations and the woman's voice felt nice to hear. Awesome!
First Last that is nice of you to say
Beautiful problem.
I wonder to how much would the angle tend to, if you kept adding squares to the right.
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
Professor Stankova is awesome! The other video (the one with the chessboard) is one of my all-time favourite Numberphile videos. More of her!
is this really something they taught in bulgarian elementary school? pretty sure i was still struggling to memorize multiplication tables back then
isnt multiplication taught in grade 1?
I'm not sure about Bulgaria, but in Poland we were done with multiplication tables by grade 3. I'd expect a problem like the one in the video as an "extra difficult" question for 6th or 7th graders.
slouch in fith grade?! That was second and beginning of third! I learned protractors in 3rd!
slouch we did this type of stuff in third and fourth grade. I’m from Texas.
Aye, bg here. Ack I remember this type of solution always looks so easy when someone else does it. Yet when I do it I have an triangle mona lisa on my list of paper and I'm trying to divide some angle by 0 to find my answer. -_-
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.
ru
I remember Prof. Stankova from the 'Pebbling a Chess Board' film. Again, she really is excellent; clear, concise & thorough.
That was neat. The actual problem, but also the sparkles in her eyes, remembering one of the things that started her fascination and love for mathematics.
I also remember multiple such situations with math, physics and programming in my youth.
This is so awesome I cried .. I'm showing this to my little sister !
how many squares do you need so the angles would add up to 180?
17 squares is 181.3. To get over 360, you'd need over 500.
Holden Watson Source?
No source needed.
Use your mind :
arctan(1) + arctan(1/2) + arctan(1/3) + ... + arctan(1/n) >= PI
It happens that n=17 squares
2. That was the most easy question.
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. :)
Welcome back! I couldn't wait for next videos with You after "pebbling a chessboard" videos!!!
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
"The size does noy matter:"
I was just looking for a comment like that ahahah it didn't take much time
Only the proportion matters 😉
If you are worried about size, just change the metric. That's what I always say.
"noy"
love the geometry segments! More please :)
Wow! Good job! Informative and easy to understand keep at numberphile
The 1988 IMO Problem no 6 killer!
Brilliant solution. Very elegant.
TheMathKid glad you liked it
Dear Lady! I love your style of presentation!!!!! What a charming experience to see it all come together in such an inviting way as you guide us around in such an expert way!!!! You are like a breath of fresh air for sure for sure!!! Thank you for this great little exercise. David Kernberger
excellent work. please make more videos on geometry questions like this.
what a brilliant teacher!
Instead of figuring Beta + 90 - Beta, you could also say that HE and HD have slopes that are opposite reciprocals of each other and are therefore perpendicular.
I love these geometric videos, please do more of them
Sumptuously. I only thought about it today, and how nice it is to see such a beautiful explanation. 🙉
"The size does not matter" -Zvezdelina Stankova2014
Time stamp please
"Exaaaactly"!! 😂
Great great great great video!
I love it.
Xalnop thank you
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.
Thanks !Happy new year!
Alright, I like this kind of Patreon reward.
Where does someone get those mega tools she's using?
What a wonderful voice. So relaxing to listen to. Please Brady more videos!
Really, really enjoy this video. Somehow missed it back in 2014!
Well suddenly, my solution involving the sine and cosine rules, and pythagoras' theorem seems horribly inelegant!
0:19 "Size does not matter", you heard it here folks.
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.
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.
This is the math people should learn at school, not the one that is filled with doing the same things over and over again.
Juju Mas it is what we learn at school though
some people need repitition to really learn and remember a concept. nothing wrong with that.
Sam Harper ah someone that gets it
Math is a beautiful subject
Bright! Time to time I see this video again, simply genius. Thanks!
I love profs. Stankova's videos. They make me feel like a kid again :)
I feel really stupid now that I resorted to trig the first time I tried to solve this!
you're not alone :D
Another way to demonstrate it
arctan(1)+arctan(1/2)+arctan(1/3) = 90°
but therefore you need the exact values of arctan(1/2) and arctan(1/3)
+Peter Auto no you don't need the exact values. Just use the tangent angle sum formula
+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
only if fifth graders knew trigonometry
Lol
That was cool. I would have never thought of doing it that way. I was focused on calculating the angles through the tangents of the triangles ACE and ADE.
Very informative, thanks for the great video! Seems simple enough to figure out the uneven shape given at bottom square gamma to top square after you see its uneven to the two beta Isosceles that are equal within the new equation given of beta + conjuncture-result - beta + something = 180 degrees, past giving its measurements at its vertical line/curvature at each original square given in the three original squares and doing the original equation of: alpha + beta + gamma = result. Very interesting, thanks for the video!
@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.
ArchimedesWorld The imagination is being reduced to almost non existence
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.
okso if there are infinitely many squares then the sum of the angles diverges
the sum of the angles formed by n squares is i/2 (log(gamma(1+i))-log(gamma(1-i))+log(gamma(1+n-i))-log(gamma(1+n+i))) which is cool
I bet it also diverges when you only use the angles going to the prime numbered squares. It feels like the harmonic series, but I don't have the math to prove it.
Ben? Ben?! It's you!
ben1996123 I like your profile picture. I also like your maths.
Here. I posted this proof in the other comment section about this topic. It shows that you can indeed use the harmonic series as minorant for the series. So the statement about the prime numbers from will ericson is correct.
"Here is a proof idea. I'm not quite sure if it's the most elegant, but I'm pretty sure it works. Also, you need first semester university calculus: We are looking for sum(1,inf)(arctan(1/n)). Define f(x) := arctan(x) - ln(1+x). Because f '(x) = 1/(1+x²) -1/(1+x) we can conclude that f '(x) >= 0 for all x in [0,1]. With the mean value theorem we get that f(x) >= 0 for all x in [0,1].* Thus arctan(1/n) >= ln(1+ 1/n) for all natural n. Because ln(1+ 1/n) = 1/n*ln((1+1/n)^n)), and ln((1+1/n)^n) -> ln(e) = 1 for n to infinity, there is a N from where ln((1+1/n)^n) > 1/2. Therefore we get the harmonic sum as divergent minorant for sum( 1/n*ln((1+1/n)^n) ) and for sum(arctan(1/n)) aswell."
*( In-detail-explanation for *: Let x be out of [0,1]. Then there exists a z out of (0,1) so that f(x) - f(0) = f '(z)*(x-0)
=> f(x) = f '(z)*x >= 0 )
Row or grid?
Approx 3:10: "Or some ugly angle nearby?" Priceless...!
Very innovative to have come up with this solution 😊
Her accent sounds so familiar to me... is she Bulgarian? (I might be completely wrong)
ya bulgarian olympiad winner
That was beautiful.
Awesome topic. Amazing solution comprehensible to a 5th grader. I hope I had access to the internet when I was in school. Numberphile, I am a big fan....
This is also my favorite math problem, and I love find different ways to solve it!
WOW we should all be thankful to the greeks for trigonometry xD
Can I use Trigonometrics here?
You get arctan(1)+arctan(2)+arctan(3)
Not clear that it sums to 90 degrees
I find this video very elegant and interesting. Thanks, Brady.
Zvezdelina, you did an awesome job leading through this proof. It felt natural and easy to follow. =)
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.
OMG!~ The most revered brown paper is cut up!~ :-o :D
I guess this is the first time the brown paper has been desecrated and destructed, or is it? :P
Vsause music starts....
I think she is one of my favorite people you feature on Numberphile! Awesome vid :)
Four years old (the video, not me) and people are still watching and commenting, which is great. Love it. Ignore the nay-sayers.
I decided to have a stab at it and was pleased to find a geometric solution with minimal extra drawing. Presumably one of the 53, but here it is anyway:
We know that we have an α at the top left in the isoceles triangle AEB. Let's try to construct a β on top of it. To do that, we need to go two units along AB, turn through 90° and go another one unit. Let's go two 'half diagonals' down to B. 45° takes us to BG and another 45° takes us to the diagonal of the middle square and one 'half diagonal' takes us exactly to its centre. I then realised that ED must also go through that centre point†, so we have AEB = α, BED = β and DEJ = γ by the 'Z' rule (EJ being parallel to AD) and AEJ = 90° because it's the corner of a square. QED
† to prove it, draw a horizontal line through the centres of the squares: it's easy to show the triangles at top top left and bottom right are congruent, but I think "by symmetry" should be enough for a recreational maths posting :-)
Omg pls why all the arctan ;-; guys this is geometry, not trigonometry. The difference is that geometry gives more of a pure proof while trigonometry is not that pure. You guys disappoint me :(
Sure, arctan(a) + arctan(b) = arctan((a+b)/(1-ab)) and therefore arctan(1/2) + arctan(1/3) = 45 exactly, but you are using a formula that i bet nearly none of the people in this comment sections, who are posting this trigonomial 'proof' here know how to prove, on top of that, proving this formula takes more effort than proving the conjecture in this video.
But if you are willing to take the time into noting that you are using a formula that has already been proven, then ok, I guess it's a proof, but it doesn't leave you with much understanding as to why the angles sum up to exactly 90 degrees.
When you are proving a geometrical conjecture using methods described in this video (congruence of shapes, equal angles and sides, right angles, isosceles triangles) you get a more pure proof which leaves you with a better understanding as to the reason why the angles sum up to 90 degrees.
On top of this all, who in the world knows about trigonometry in freaking fifth grade, i mean come on. If you walk up to the most clever fifth grader you know and tell them about your trigonomial 'proof', do you really think they understand? I think the geometrical proof will be way easier to get your head around as a fifth grader.
And to those who are just saying 'arctan(1/1) + arctan(1/2) + arctan(1/3) = 90. There. Done', just go away and enjoy your oh so open minded life.
Secondary school level okay... even this way of solution
That blew my mind mentally and physically. Amazing
Proving space using Negative Space. I've never seen that applied to angles, but it's blowing my mind. And I love it.
USING ARCTAN IS NOT A PROOF UNLESS YOU USE A FORMULA TO SHOW IT ANALYTICALLY. ARRRGH
cursedswordsman Indeed, it seems half of the people here think that taking arctan on their calculator is fine.
cursedswordsman You are wrong my Little friend
Hue hue arctan :P
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.
+Harry Potter granted I'm a chemist so I believe integral are like addition lol
So a fifth grader is supposed to know all that ?? :O
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.
Love it - more from her please!
excited to have stankova as a professor next semester!