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MATH • GEOMETRY • The Bermuda Triangle and the Mysterious Square
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The Bermuda Triangle is a place in the Atlantic Ocean where ships and airplanes supposedly disappear without a trace. In the picture below, a square appears when we rearrange the pieces of the upper triangle to form the lower triangle. The pieces in both pictures are identical. Can you explain the origin of the square? You will need your knowledge of geometry to solve this problem.
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149 comments
 
Try compare them in Photoshop, you will see the difference:)
 
The slopes of the two triangles are different... ;)
 
The hypotenuse isn't a straight line. If you examine at the slopes of the lines on the green and yellow triangles, you can tell that they are different.
 
Wow. I've seen this 'puzzle' before and never figured out the answer.
 
I couldn't figure out.........
 
@Marco Amersfoort, look it up. the answer is in google. the yellow triangle has a arch in it that takes up area not seen with the eye. fun puzzle.
 
Had solved this during my first year of engineering... Took full half day to solve this... Very good puzzle
 
Yeah, I saw that. There's a link to the answer right above the image. That's why I was amazed as I was :-)
 
it's a fake ! the yellow 1 it's ittler on bottom ! XD
 
yeah, I'd seen this before. +Brandner Kaspar and +Paul Schuler have it correct. It's pretty clear that the slope of the green triangle is 2/5 (0.4) and the slope of the yellow triangle is 3/8 (0.375). So what you actually have is, in one case a slight concavity and in the other, a slight convexity, but nearly indistinguishable to the naked eye.
 
I don't get why this is confusing. It's the same reason you can't repackage merchandise after you've taken it out of the box. You're putting the same things back but since they aren't arranged the same way, it seems impossible to make it fit in its original box. It's all about arrangement.
 
@MerzC SVoRN, yes I agree FAKE! all you need to do is change the size around right. lol everything on the internet is fake. can't believe what you read or see in here.
 
the gold and green triangles do not have the same slope (green => 5/2 = 2.5 , gold => 8/3 = 2.666). Therefore, if you interchange their positions, you must make up for the discrepancy with an extra square.
 
For more fun, look up vanishing leprechaun... Similar trickery...

Enjoy!
 
The only problem with this is that the slopes are not identical.
 
Examine the slope. I have used that in a class for years now.
 
2/5 != 3/8 ... different slopes ...
 
Brilliant deduction +Thomas Parker ! You deserve a cookie! Yes you do. Yes you do! Dawww, look at how cute he is trying to solve grown up problems.
 
The pictures are the same. Even counted the boxes.
 
OMG! The answer is great. Ofcourse the new white box increased the area. How could I not understand it at first?
 
Area as a sum of the parts: 5 + 12 + 7 + 8 = 32. Area of large triange if all lines were straight: 32.5
 
it's just that not noticeable, damn it (ok i peeked at the answer).. lol
i was just like, what the heck is wrong with this? my eyes deceived me.
 
first let me tell u that i am a child not grow or a teenager do u understand....
 
i solved it in 20 min. the main geometry is not triangle because of the different slop of green and yellow triangle. area of upper geometry=32 and lower=33
 
maveric, y do u care about geometry
 
Perfect timing. I was trying to explain this to someone offline a couple of days ago. Just emailed the link to them. Thanks!
 
dont get it :/ DONT BE MEAN IM NOT GOOD AT MATH!
 
dont fell bad danielle... i am worse at math than you
 
i am so confused... stop making me confused, lol
 
Ugh, this again? The top one is not a true triangle.
 
if you want to be in my circles say i
 
if u want to be in my circle say pie
 
who wants to be in my circles
 
Lets have a math question where we call something that isn't a triangle a triangle and then claim a paradox
 
noo you r not and u never will be
lol
 
pi, pie, me oh my, we all need pi
 
when you go in the square you dont dissapear! :-)
 
maveric, i added you to my circles
 
hell ya hundreth comment
 
brb peeps, mavic add me back soo we can have a private convosation... brb peeps, and u r not tom cruise
 
wooops spelled yo name wrong but brb
 
This is absolutely retarded. It's an optical illusion. Look at the lines. They have been distorted slightly in order to seem like they're the same right triangle. They're not, though.

If I could -1 this idiotic post (and page), I would.
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I'm a kid so, if you are looking for an answer, don't look at me
 
That's where the worm hole is located which is where people get lost at
 
hey... the star has returnd
 
first thing that came into mind when i these grids: it looks like two parrot heads wearing blue sunglasses. lol. inkblot thinking conditioning. *yawn*
 
This appeared in Scientific American sometime in the late 50's. It looks impossible until you examine the construction very closely.
 
tch its elementary watson
 
I AGREE WITH BOBBI whoever you are
 
False. The shapes are NOT identical. Mathematics proves that no matter how you arrange various shapes, they take up the same amount of surface area. No more, no less.
Also, This is not an accurate representation of the Bermuda Trianlge.
 
I don't understand jack shit about this. All I know is don't go in there unless you wanna commit suicide.
 
another you r just moving the squares and shapes around
 
its a trick, even though the top line of the triangle looks straight it isn't. go get a straight edge and see for yourself
 
it has to do with area and perimeter, for instance a perfect square has more area than a rhombus though they may have the same perimeter IDK???
 
No matter how you arrange the pieces (even when the square appears) it will have the same surface area. Just take some squares and arrange them in a different way but the area doesn't change. It is not that mysterious. Just an illusion, if I say so myself. And look up at +paul stone 's comment. Both are logical explanations.
 
both do have the same area but the top line's not straight, check :)
 
or... the angles of the green and the yellow triangle look the same but aren't
 
There is something wrong with your conclusion here because obviously the blank space in the bottom triangle would be a glaring oversight if you are trying to say that they occupy the same area.

Perhaps this was just a mistake in the graphic but I'm sure I'm not the first one to point this out. Maybe I'm just the next one in line.
 
put a straight edge along the green and yellow triangle, trust me
 
this is getting as old as the hills.... 3rd time in Hot on Google+?
 
The fact is these two are NOT triangles. Nuff said :)
 
The puzzle is incorrectly posed because it lies by referring to them as triangles when they are not. A better worded puzzle would refer to the top and bottom diagrams because then the assumption of a triangle is entirely the viewer's fault and the lesson not to make assumptions is more sharply focused.
 
+Brandner Kaspar : Well I said the two big ones are not triangles (if they are, then 2/5 = 3/8 = 5/13, which are not)
If you understand what I mean, then you'll know that these two are not triangles and are not the same. The one on the bottom is a bit bigger, hence the white space :)
 
nice one, the slopes of the two smaller triangles are different. one has 2 on the vertical side and 5 on the horizontal. the other is 3x8. these two aren't equal slopes and hence those two smaller triangles cannot be aligned in one line. The big triangle is really not a triangle. it is an illusion!
 
the surface area is different in each unit and the optical effect is a confusing comparison of similar shapes.
Zee Dee
 
It looks like it would make for a really boring game of tetris.
 
Try again cuttin pieces of paper ;)
 
The slope oh the 2 triangles (green and yellow) are different. So its not an actual a triangle.
 
Very cool! I love this kind of math!
 
8/3 not = 5/2
i.e. the "hypoteneuse" of the first "triangle" is not a straight line.
 
What appears to be the hypotenuse of the overall "triangle" is actually two line segments. They form a convex bulge in the overall shape when the yellow triangle is on top and an concave indentation when the green triangle is on top. An outline around all four intersections of each of the resulting overall figures reveals two quadrilaterals; one with an area of 33 squares and one of 32.
 
Very cool. Geometry problems are still fun and interesting.
 
I didn't figure it out by myself
 
The only thing that the bermuda triangle has to do with geometry is the name.
 
as i observed, the above yellow triangle is not of the same size as the yellow triangle below. same with the green triangle
 
+Aidgur Dinny the small triangles absolutely are the same size in both cases. And they absolutely are real triangles. The overall shape you make by putting them together is not (quite) a triangle however. I suggest that anyone who wants to understand this a bit more gets some gridded paper, a ruler and scissors, and reproduces this illusion in real life. I can be done. When I first saw this, that's what I did, and now I really understand it. Measure and cut out the four separate shapes carefully, and rearrange them yourself.

 
the gold and green triangles do not have the same slope.
 
The real surprise is in how much area can be lost in such a small variation in where the indentation of the hypotenuse lies.
 
The slope Of the anglesl When you changed in any direction Buy a degree or even less would Be enough to leave a positive or negative space, Mostly in the yellow is a about 85 percent and the green about 15 percent not really noticeable by looking at it right off the bat..Dam..i sound like some friggin science geek, I'm a body piercer and a tattoo artist, I just set up this igoogle mail thinger and I don't even know why it was sent to me or how this thing works at all....MARCEL
 
Ya I figured. I had a problem with the base by height ratio's not being the same.
 
Это оптический обман, в одном случае треугольник одна из сторон выпуклая, а другой треугольник вогнутая)
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Shenanigans!
Green triangle slope: 2/5
Yellow triangle slope: 3/8
 
The human eye is amazing, but it can't see everything.
 
another great the words. i find since visually impaired. chemo. thanks to technology , no longer portrait artist, yet the graphics and camer is my eyes....Ali comment was such. love the balance. thank you...
 
It is impossible all 3 sides cannot be straight in one or the other or both as it does not add up that way as displayed. The photo does not show enough detail.
 
This one is well known, the upper one is slightly smaller then the one on the bottom, but we can hardly see the difference, but the slight difference is distributed on all the diagonal faces and if we accumulate it makes 1 full block
 
the top line does not cut through the background arrays in the same exact position when you look closely...that's the trick
 
The simplest explanation I found is that 2/5 is not equal 3/8 so... the top big thing is not a triangle.
 
the answer is in the question, i.e. angle
 
Two triangle are not same tangent.
 
lol they have the same area the small square appears because the first figure is a quadrilateral, the hypotenuse in the first and the second "figure" is not a rect... check it closely xD
 
i dont think the hypotenuses r straight.........
 
were not playin tetris here people
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