The Three Square Geometry Problem - Numberphile

So what are we talking about today? About my favourite problem in geometry when I was a fifth grader and it is etched in my memory to the point of being able to recall it at any time so we have three squares of the same sizes next to each other the size does not matter so what we will do you is pick one of the top corners let's say the top left corner and connect it to the bottom right corners of each of those three squares. Next, we will take this angle here at the bottom let's call it alpha the next one at the bottom let's call it well beta and and that's incorrect yeah this is upside down and finally gamma alright so the question is asking what is the sum of these three angle so it's a simple enough problem the interesting part about it is that it was given to me as a fifth grader so I had to a attack it as fifth-grader in fifth grade we learn how to use a protractor so I measure the first angle it's about 45 degrees and in fact then you can stop and think about it but of course it has to be 45 since the triangle we're looking at is right and iscoclese this guy here so it should not be a surprise to get a 45 degree angle there and we go about measuring the others there perhaps we run a little bit out of luck because the second angle looks like it's pointing to 25 and a little bit maybe 26 and the third angle Wow where is it going perhaps 15 20 so about 18 so when we add this up 89 could that be it what do you think it seems a bit arbitrary i don't know Yeah and also there were all of these mistakes in our measurement and why would they be asking a fifth grader how much this is and ending up with such a random angle? well if you try to make another measurement perhaps you will come up with 91.3 and so you would say oh it's a little bit obtuse but maybe it's a little bit acute so what would be conjecture that people will leap to if they believe this is a beautiful problem I would guess that the correct answer they would think the correct answer is ninety exactly ninety but how would we know really is it ninety? Or some ugly angle nearby? we cheat a little bit we say okay that problem was given to a fifth grader it has got to be a nice answer we will believe for a moment it's ninety and this is what mathematicians call a conjecture and then put all of our effort into proving that it is ninety and we are kind of a third of the way to proving it because we know that one of the angles for sure is forty five so what should be left for the other two angles in order to make ninety? well obviously forty five. so somehow we need to add up the other two angles beta and gamma and prove that they add up to up to forty five another experiment which a fifth grader might try and you can also do at home is cut up the whole thing with scissors and eliminate some of the error that is coming from the protractor when you are rounding the degrees okay so let's try to put everything together if we can it does look like this is a ninety degree let's check well yep it does look like 90 degrees okay so now we truly believe the statement and we want to prove it well the question is at what level of your mathematical career or mathematical life you are because this problem has not only one not only two but at least 54 solutions. 54 solutions? No kidding. Are we gonna do all of them? No. we will do the most brilliant one the most simple one that a fifth grader may be able to grasp so the idea of our solution at least the attempt when we cut with scissors can be carried forward mathematically so we may want to move all three angles into the same geometric location in our picture so we need to find the best geometric location where to move those three angles well there is no best geometric location until you know what the solution is so we might as well just try to move them all the way in this corner next to gamma because we already have a right angle here this guy we have gamma here and so the other two angles alpha and beta must somehow fit and here comes the brilliant idea for the solution as I a said there is something missing in this picture what's missing is half of the picture we will replicate these 3 squares up and then we'll start seeing the truth and now i think I'm going to start labeling things so people can refer to them later on I'm just labelling all of the important points I will choose point H and connect it to two other points D and E and I will say that no more constructions will be done in this solution we somehow have to make sense of what this picture is telling us well for starters we see 3 angles exactly where we wanted them to be where this ninety degree angle was we have our original gamma and then 2 more could those two angles be indeed alpha and beta well you can kinda guess that this angle is bigger than the next one by the way it looks so perhaps that will be the 45 alpha because alpha was bigger than beta could this be forty five and let's try as detectives figure out where it could come from this picture it is involved in this large triangle did we see a similar situation earlier in the problem certainly right here where we saw our original alpha and where we decided that alpha is 45 degrees because this was ABE was a right isosceles triangle do we have a right isosceles triangle her we might could EH be equal to HD these are the two extra segments that we draw on top the answer is yes because these two segments are hypotenuses Hypoteni or hypotenuses? so EH and HD the two segments which we drew are the hypotenuses of two right triangles two that we also draw on top so I'm going to mark them in blue so these two sides are hypotenuses of two right triangles E F H and H I D the ones which I drew in blue now they are right because they have a right angle here coming from top but they are also exactly the same size and shape this right triangle has one leg two and the other leg 1 and this one also has a leg of two and leg of 1 which means that their third size, the two hypotenuses must be the same so indeed EH is equal to HD and we are on our way to show that this big green triangle in the middle is right isosceles it's already isoseles but the question is why is this guy right okay but what else is around this angle there is this guy here this angle in this other one what do they add up to and the answer is also very simple one of thise angles we already know I claim this angle is our angle beta because there is a third triangle which is precisely the same shape and size as those because it is right here again a right triangle with legs 1 and two and so therefore this angle beta must be equal to this single beta and guess what it must be equal to this angle beta they all share exactly the same angle so accidentaly we have shown that we have moved angle beta to where we wanted it gamma is there and if that is really 45 we will be done and again backtracking we wanted to show that this big triangle was right isosceles we already know that this angle here is beta what is this guy well this guy this second angle participates in a right triangle where the other acute angle is beta what could this guy be we know that the angles in a triangle add up to 180 this guy is ninety therefore these 2 must add up to 90 simple arithmetic shows this angle here must be ninety minus beta ninety minus beta something and beta must overall give you 180 what is the remaining angle here this curly one if you start adding up let's do it beta plus ninety minus beta plus something must be 180 so beta goes away and that's something turns out to the our 90 degrees you've done it! that's right so our angle alpha is indeed moved here along with beta and we can conclude the the three angles add up to 90 degrees and this is an amazing construction which uses only squares isosceles triangles in congruent right triangle with legs 1 and 2 and that is indeed a brilliant approach which mathematicians take to solve problems they complete it to what's missing there and then the problem becomes easier

Channel: Numberphile

Views: 923,604

Rating: 4.8701739 out of 5

Keywords: numberphile, square, triangle, Geometry (Field Of Study)

Id: m5evLoL0xwg

Length: 12min 21sec (741 seconds)

Published: Thu Sep 18 2014

Reddit Comments

[u/Vitrivius]

She sounds like Dexter from Dexter's Laboratory! I love the accent.

[u/demultiplexer]

Videos like this make me long for the days in which I got straight As in geometry :D

I loved doing stuff like this.

[u/Loosner]

If I were given this problem, I would just set the squares sides equal to 1 and calculate the angles, and add them up. I am one of the few people who loved algebra and calculus and hated geometry. Why is that? I guess that is why I chose engineering instead of mathematics.

[u/XI_Ki11JoY_IX]

I think the better question is:

If arctan of (1/1) or 45 + , the arctan of (1/2) or 26.56505118... + the arc tan of (1/3) or 18.434994882... = 90 , what is tan of 90? ;)

[u/Alienturnedhuman]

I came up with a simpler proof for the puzzle, which I filmed and put here:

https://www.youtube.com/watch?v=DCUGTyz61-g

I don't know if it is one of the 54 proofs already known about or not. I mean, I would assume someone already found it, however the implication was that the other proofs were far more complicated, yet the proof I came up with is simpler than the one shown on Numberphile.

Unless, of course, I've made an error, so I welcome feedback and discussion!

[u/tomtomgo314]

Paused the video at 5:10 and worked on this for an hour. Heres my solution video... https://www.youtube.com/watch?v=BxUtMfaPuRs&feature=youtu.be