How can this be true? Observe the picture precisely, and answer will come. Feel free to comment this illusion, but please don’t spoil other’s fun by revealing the answer.
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Wow, all this pointless math equations and various theories about how they aren’t true triangles. You’re all looking at it too deeply, take a step back, will you?
It took me a total of a minute to figure out what was going on here.
It doesn’t lose it’s spacial area in the second picture because:
1.)The red triangle’s height more than makes up for the loss of the orange non-triangle shape.
2.)Although the horizontal length of the green triangle isn’t the same as that of the red triangle, this does not matter because the movement of the orange non-triangle makes up for it.
It’s just simple movement of objects people. No real optical illusion going on here.
-Paul
You people are way to into it.
Just think, the only other way to make that triangle other than the first way, is the put the orange and green parts like they are.
The orange has 3 squares until it has another layer thing(the 2 squares ontop of 2 squares).
The green has 2 squares until is meets another layer. That’s where the gap is.
**Think of tetris.**
hello!!!didnt you see the columns one of it is big and rectangular!!!!! making you count 9 and the other 8!!
Well, I just tried cuting out the pieces myself, and noticed that you have to draw the large figure first and then draw the little pieces inside. The pieces, as dimensioned int he first figure, do not fit correctly into the large figure. I’m not going ot argue, but if you want proof, just try this:
Draw a 5 x 13 right triangle.
Make a 3 x 8 and 5 x 2 right triangle, as well as the yellow and green pieces to specification.
Try putting the individual pieces you made into the big figure. You will hopefully notice that they don’t fit, wither way you try to put them in.
And for all the people who just looked at the “levels” “arrangement”, or “number of boxes”, how do explain the fact that both of the large figures are the same dimensions, and made of the same dimensioned pieces, but a chunk is missing? Area doesn’t dissapear…
Does this mean I have to read the comments including math in them?! I saw this on metacafe.com. Check out their illusions. Cool! One of them teaches you how to draw illusions. I tried it, works!
Okay, I may be an idiot, but this isn’t making sense to me. The curved line explanations aren’t relevant to this illusion. There are no curved lines. If the lines were curved we would be able to see it. People wouldn’t be able to construct their own puzzle out of graph paper and arrive with the same missing square.
I know it has something to do with positive and negative. Like the riddle about the money that was split three ways and when it was returned, they were a dollar short. When it comes to this puzzle though, I don’t get it. Can anyone give a simple clear explanation that doesn’t involve curved lines or stretched triangles?
Wow. All of you people who try to come in here and sound SO smart talking about curved edges and concave and convex lines and whatnot. Instead of trying to be all high and mighty…why dont you make one out of paper and see how it actually works, it took me two minutes to make one (using a straight edge, NO CURVES) and as soon as i had made it the answer became obvious. The fact is, in this situation we arent talking about the area of the triangle, what you need to consider is the size of the edges of each object. If you go and make one out of paper then rearrange them inside the hole you cut it out of you can see that both shapes have the same area. It is interesting, but what happens is, because the larger triangle is 3 high and the smaller is 2 high you have to stretch out the two L-shaped pieces, instead of leaving them lying on top of each other, so that the two of them will equal the length of the large triangle’s bottom side. Its hard to explain exactly how it works, all I can really say is if you still think it has something to do with the triangles being curved or different sizes or something, its none of that, just make one and you will see. Its very easy. And PLEASE people…don’t try to sound so high and mighty all the time, just makes you seem REALLY dumb when you end up so wrong.
Ummm… I kind of understand it. I don’t get all of the “2*5/2=5″ or the “3*8/2=12″ and the other stuff. And to all of yous math people with the curved lines, maybe the person drawing it made the red and green pieces a bit off by accident.sheesh.
=|
Thank you, Dizee. This is how it works…Rearranging the pieces like that causes the hole because of the two L-shaped pieces. Notice that the long part of the orange one is longer than that of the light green one. The triangles are there to compensate for the space…don’t get me wrong, I’m not talking about area, I only mean that the triangles could not stay in their original positions. Thank you.
For everyone that has tried to explain the convex,concave fact, good job.
For everyone else that doesn’t believe it’s that simple, sorry, but it is.
Let me try to shed a little more light. In order for this to be a true, “straight” triangle, any point along the diagonal should have the same slope (remember “rise over run” from geometry?) The whole triangle has a slope of 5/13, but the red one is 3/8. These are not equal, but they necessarily MUST be if the triangle were proper. Additionally, the green triangle has a slope of 2/5 which also isn’t equal.
If you’ve tried to construct this on graph paper, no doubt you counted the squares the same way and therefore have “introduced” the necessary flaw into the construction. That’s why it works.
y do people think this is hard…maybe it only makes sense to asians like me
this is soooooooooooooooooooooooo easy CANADA ROX MONKEY SOX
I still dont understand how it being a quadralateral makes any difference… but I’m not amazingly clever and I hate geometry, so…
How can you not get this?? The portions are NOT THE SAME for the red and dk.green parts. They cut off some parts. Those parts all equal one more square that fills in the spot. I didn’t use photoshop or anything? I just used ym 6th Grade mind!!
Look at the slope of the two triangles. (For those who don’t know, slope is calculated as rise over run) The slope of the green is 2/5. The slope of the red triangle is 3/8. Now the difference between the two is very small, and in the picture, it is hardly noticable, but those who said that it is a 4 sided object are correct
hint:
are you sure they are two triangles.
i saw this puzzle over a year ago, and aven as a 15 year old it was simple to me, i immediatly saw a difference in ratios between the verticle heights and the lengths if the traingles, being 3:8 and 2:5 which must therefore for a 2 lines when connected, not the apparant 1. It is this difference in the angles of the 2 lines, being an outward angle in the bottom “triangle” (not no more), which adds th area of 1 square to the shape, so the area of 1 square can be removed from the bottom.
Clever, but not clever enough im afraid.
I even beat my maths teacher at this too
so thanks
It’s funny to me that some of the people responding “oh it’s so easy, you’re all dumb” are wrong.
The figures made by the 4 shapes are NOT triangles, they are quadralaterials.
Make a transparency (as someone earlier suggested) is the easiest way.. and doing it on a computer is best because you can make sure you are 100% accurate.
Of course, you can do it mathematically, but those that would think to do it that way wouldn’t be suckered in by ‘assuming’ similarity or congruency, would you?
First, if anyone actually reads down this far, like I did… wow, you have too much time on your hands.
Anyways, It’s been explained right multiple times, but not to the point of where a, say, 10 year old can do it…
SO. This site explains it perfectly, and illustrates it so you can see it.
http://www.scientificpsychic.com/mind/triangle1.html
You see, the two full “triangles” aren’t real triangles, BUT THE SMALLER TWO ARE. If you cut it out and put it together you WILL get the same answer. BUT: try it with these demensions and you’ll see the illusion alot better: The top triangle has length 3*5, and the bottom one gets 2*1. The other shapes become 2*1 and 1*1 rectangles. When you transform it, switch the triangles the same way, and put the larger rectangle above the smaller one. You’ll REALLY see the difference.
Or? Just draw a line from the very top corner of the first big triangle to the left-most corner in photoshop, then zoom in on the point where the two triangles meet untill you are bored.
If what I said doesn’t make sense… then you must not know what either a triangle, a line, or a rectangle is…
plain and symple trigonometry and geometry puzzle…. puzzle i said?
as easy as this:
it is not the same a square area inside 2 by 8 pices (16) than a square area inside a 3 by 5 pices (15)
If you sum the pices of both non triangles you get 15 but if you rearenge the pices like in the puzzle, you have to fill a 16 pices area, the triangles are there to confuse you, their hypotenuses can make a perfect straight line while variating the remainig space that has to filled eighter with 16 or 15 pices is made by the opposite and adjacent legs of the triangles. So it can be a space made of 5 by 3, or a space made of 8 by 2…
you just have to forget about the triangles and focus on the remaining space designated by their opposite and adjacent sides…
Ho Drakon, Ho Megas
Simply put, The rise of the red and green triangles are not the same. Green has a steeper rise than red (saw this in photoshop and with cutouts). If you look carefully you can even see that the bottom and left side of green is 5 by slightly more than 2 and red in the same area is 5 by slightly less than two. Never took geometry but obviously the extra space is to compensate for the greater rise of the green trangle. Try it and you’ll see. Cut out both trangles and simply line up the points. You’ll see that green has a steeper rise than red. Hope this doesn’t sound stupid. I tried.
ooh i know this one! The triangles are not exactly the same shape. The top one has bent edges but they’re really hard to see.
I saw this on some web page with the proper explanation but I dont remember where… hehe, sorry.
Easy. I will not spoil it though.
ok woooww it definitley has NOTHING to do with bent sides…in the top triange, if u take out the two ‘L’ shaped objects, u have an empty 5 by 3 rectangle which means the are is 15. In the bottom triangle, if u take out the ‘L’ shaped objects, u have an empty 8 by 2 rectangle which means the area is 16 and there is one more square then the top triangle.
idk how that works but thats all i could think of
Find the tangent (tan) of an angle A in a right angled triangle with side 1(opp) and 4(adj). The hypotenuse is the side of the triangle opposite the right angle. The opposite side is the side of the triangle opposite A. The adjacent side is the third side. To get Tan(A) divide the length of the opposite side by the adjacent side. So in this triangle, Tan(A) = 1/4.
A triangle with Tan(A) = 1/4
For example, the steepness of a road is often shown on road signs as a percentage. If the road surface is at an angle A away from the horizontal, the steepness percentage is 100*Tan(A). What steepness percentage is a road at 45 degrees? How about 63.4 degrees?
I could give u d answer but dats not as fun !!!
the answer is easy. The 4th column from the left is larger than the other ones. so the area isnt the same for the red one in the top triangle as it is in the bottom one.
*****SPOLIER ALERT*****
the real trick is in the smaller red triangles go back and count how many squares are in them. you will find the top one has one more
ohhhh haha,
duh.
i get it :)
I don’t know if people will read this, but here’s the explanation to why it works when you make your own triange.
I agree with all the people who say that it has to do with a bent line. That is correct. The people proved this with easy math.
So what happens when you use all straight lines, you ask. All that happens is that the second triangle is bent outward even more that it would be if you originally started out with a concave line. The difference in area between the triangle with straight lines (the first one) and the triangle with the convex line (the second one) will still amount to one square.
The reason that the makers don’t start with a straight line for the first triangle is because it makes the second bulge less obvious by spreading it out between the two triangles.
I hope this clears things up with people saying that they made their own triangle and it worked.
p.s. And just in case you didn’t get it before, the triangles I am talking about in my explanation all refer to the big triangles, not the two small ones that make up a larger one.
The people laughing like idiots and saying it has nothing to do with bent lines fail to explain why the bent lines EXIST.
Of COURSE it has to do with the bent lines, otherwise the lines would not be bent in the first place.
uh, öh uudndfnbfkjhdf!!!! I MADE IT FROM PAPER!!! eefeeudyd!!! OMG!! OMFG!! ZOMG!! I CANNOT UNDERSTAND IT!!! EEEEHakkaöaöä!!
HUH????
the red one is the biggest so u have to move the green one back a space it took me 10 seconds to figure it out and i’m 13
***Spoiler Alert***
They have different areas because the hypotenuse (top left line) of both triangles are not actually lines. They are to separate lines which inter sect at an angle just above or below 180 degrees (depending on which triangle). Since the lower figure has a bigger angle it takes up more space and therefore leaves no room for the missing block.
This is how I know that the two hypotenuses are not actually lines:
In order for a line to be a line, it has to have the same slope throughout the whole line.
If Slope is m
then
m = Height / Length
The segment of the red triangle that makes up the supposed top-left line goes up 3 BLOCKS and across 8 BLOCKS so therefore has a slope of “3/8″
The segment of the turqoise triangle that makes up the supposed top-left line goes up
2 BLOCKS and over 5 BLOCKS and therefore has a slope of “2/5″
Since the two line segments have different slopes, they can’t possibly make up a single line segment. Therefore, the two big triangles are not triangles afterall, but are actually quadrillaterals (4 sided polygons).
Wow its easy the area of wich the blocks were placed is diffrent so it messes it up…. well its hard to explaine…. and whats with the pointless math its just a frikin elousin!!!!!!
If you have a curved line that takes up space and then you take away that curvature it doesn’t produce a square out of thin air. The area of one square is either contained or not contained within that curve. Especially if the damn things are not pressing against any other objects and just hanging out on top. I don’t have an answer to the puzzle but that’s my response to the curvature theory. BS!
You can tell the first is not a triangle because of he slopes do not add up
here’s the answer:
http://www.flickr.com/photos/nickleus/2527301881/
ohhh my god nr 5 is right all u other smart ass’s diden’t get the point, of the Illusion, im lauging my ass of haha, u think u are so smart yet u are so dum,,,, just look at the link at the 89..
Use the grid as if it was a graph. You would see that the slope of the bigger triangle is 0.375 and the slope of the smaller triangle is .02, which indeed is a big difference. That means both the lines on both the triangles are curved. The first “whole” triangle is concave (going inwards) and that makes the triangle look plausible. The second triangle is slightly convex (curving outwards) which is hard for the human eye to notice. That little difference is actually equal to the missing square’s area. so the curved lines are the whole idea of the illusion.
If you use a very accurate way to measure this you would see that what I am saying is true.
I’ve figures it out
The Top triangle goes up by 5 squares
And ythe bottom triangle is 6 squares but it tries to trick the human mind by adding a rectangle istead of a square but they’re
5:6 not equal
The area of two triangular pictures is not same.
Red triangle area in both pictures = 3*8/2 = 12
Green triangle area in both pictures = 2*5/2 = 5
Area of two blocks in top picture = 3*5 = 15
Area of two blocks in bottom picture = 8 + 7 + 1 = 16 (each smal box is considered as 1×1)
The two pictures look equal but they are not equal. Zoom and look at the 6,3 from bottom left on bottom picture and then compare the same location on top picture.
For those of you who don’t understand geometry, try this:
Cut out the all shapes TWICE.
Arrange the shapes like the two sets above then put one whole set on top of the other.
Are they the same size?
You should be able to see that the longest side is different in each set.
This is why, try this:
Put the red triangle on top of the green triangle so the pointiest parts are together.
Are they the same angle?
You should be able to see that the angles are different.
Now, if you can’t understand that, then we need to introduce an IQ based population cull …
Honestly, this whole thing has nothing to do with curves.
The curves are just slight errors. The real mathematical phenomenon occurs with the orange and green non-triangles.
The area of these two pieces will never change. But by moving the bluish triangle to the bottom row and the red triangle to the top, the perimeter of the orange and green non-triangles, combined, is forced to change. If a shape’s perimeter is altered and is not allowed to change area, a gap is formed.
That explains the gap, but I am still unable to come up with the larger question, “Why aren’t the areas of the two triangles the same?”
I repeat: There is no trick in the curves, just cut the damn shapes out yourself using a straight edge.
1. Two Non-Triangle shape dont make any difference. This streching-out thing which has been explained by some posters is pure S***. They are more adept in Tetris rather than Geometry.
2. As explained earlier, difference is because of Concave and Convex Hypotenuse ( Calling it so for namesake.) It actually IS NOT a single line.
Mathematical proof has been given in above posts. I'll just make it simple for those who are making it on graph paper.
> Draw TWO sets of the above image. NOT in single piece as BIG triangle, but with 4 given small figures.(precisely to scale)
> Arrange set ONE to form top triangle.
> Arrange set TWO to form bottom triangle.
> Place the FIRST formed triangle on top of SECOND formed set.
> Notice the INCONGRUOUSNESS in HYPOTENUSE.
> Have some belief in yourself. This is not because you made some mistake in cutting. This is there because a DIFFERENCE IS THERE.
———
I cannot state this in simpler form than this. If you still haven't got it, Enjoy the mystery and don't try to solve it any further.
i figured it out realy easily
it is realy quite impossible
I figured this out in a minute, But my dumb ass friends still don’t believe me!!! so they cut out peaces of paper and tried to prove me wrong. Oddly enough because it is so minuet they “proved” me wrong until i told them to fit it into the “triangle” hole they cut it out of.
Like said before, simple geometry.
it is like some of you retards want to believe that this is ‘magic’.
those are not actually triangles by definition, so the area of the two using 1/2*b*h is irrelevant. plus they are obviously two different shapes so you wouldn’t be able to use the same formula to calculate area anyways…
most people should be able to understand that since the two actual triangles don’t have the same angles, the two hypotenuses connected can NOT make a line, but can only make two line segments…
but for you retards- pick a vertical grid line and look where is crosses the ‘hypotenuse’ of each of the larger ‘triangles’. notice it isn’t at the same horizontal space. ergo, they aren’t the same shapes.
The illusion is simple. The blue triangle is NOT the same in both images. In the original figure the triangle is 5 x 2, while in the bottom figure, it isn’t. How do I know? The slope of the whole big triangle is 5/13. That can’t be reduced, which means that the little blue triangle CANNOT be actually 5 x 2. It’s really about 5 x 1.92, and that small little error, although undetectable to the naked eye, makes up that last open square.
NO MORE COMMENT HERE IS THE ANSWER:
http://www.youtube.com/watch?v=9UjYwUkbhcU&feature=related
the bent or even the degree is not the answer it is just the area is the illusion because it is almost impossible to see when your not in focus. the drawing their you see is wrong… just see my link to prove it.
here it is
http://www.youtube.com/watch?v=9UjYwUkbhcU&feature=related