By Vurdlak on March 19, 2006, with 191 Comments
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.
(77 votes) (77 votes)Comments
191 Responses
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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
It’s totally as Jiggs up said the interesting part of all is that the RHOMBOID that is formed, is exactly 1 square unit. We use this examples to optimize the areas when designing in architecture, cause it’s a minimum space lost y long distances, then turned into useful square spaces. –Fernando Barragán–Mexico City.
To Do’h: first of all, it should be spelled “D’oh.” Also, as far as I’m concerned, velocity has to do with the rate at which an object moves.
**Spoiler**
Comment ten is correct. I don’t really care about it not being a 180 degree angle. What happened is that instead of putting the orange L on top of the green L, it was moved over one square. Since the red triangle is one unit longer than the dark green triangle, it was able to fit.
Paula: good luck in geometry class. I hate geometry, but if I pass this year, I don’t have to do it again in high school.
look, the triangles have different angles- since tan (red) = 3/8=0,375 and tan (blue) = 2/5=0,4
that means that both shapes aren’t triangles, but polygons with a blunt corner at the point where red and blue meet.
so how did we learn to calculate the area of a polygon in primary school? draw a rectangle around it, calculate it’s area and subtrack the area’s of the triangles and rectangles that surround the polygon. (it’s hard to explain for me in english but try to imagine/remember it)
well, the area of the rectangle is 5*13=65 and the area of the first polygon is therefore 65-33=32 [65-(5*2/5 + 3*8/2 + 8*2)]
the area of the lower polygon is 65-32=33 [65-(5*2/5 + 3*8/2 + 5*3)]
the area’s of the coloured shapes remained the same in both polygons, but this illusion makes you think that both supposed big “triangles” have the same area. but in reality they’re two polygons and the area of the last one is 1 square bigger than the first (33-32) and that is created by the empty square you see.
i didn’t really say any new things, but i’m just trying to make it a little more clear than in #10
I’m sorry, I meant #5 in the last bit, what #10 said was ridiculous… all the credits to jiggs up
ah ha! I see it now. high school trig saves the world again
Open Photoshop, then start a new project on a perfectly square canvas. Now show your grid guidelines (ctrl+’). You will need to go into the gridline preferences and make the gridlines for every 20%, with 3 subdivisions each. This will display a grid of 15×15 squares.
The Triangle is 13 squares wide, and 5 high. Draw just one triangle with these dimensions, photoshop will automatically snap to the corners of the grid boxes. You have a perfect triangle. Now realize the dimensions of the orange pattern. It’s supposed to be 5 squares across on the top row, and two squares on the bottom left, occupying 7 of the grid boxes, perfectly. Look at the triangle you drew in photoshop. The top left corner of that orange shape, will not fit cleanly between the red and green triangle, no matter which configuration you have the partitions set. The top left corner of the orange shape would have to meet exactly at a corner of one of the grid squares. But as you can see by the perfect triangle you drew, there is not one point within the hypotenuse that crosses a grid box exactly at the corner. So the Orange segment will not fit as the illusion suggests. In the illusion, the hypotenuse crosses two exact corners of two different grid boxes. This is not so when doing so on a mathematically true grid and perfect triangle. The illustration here is an illusion, the red and green triangles have to cheat in order to cross grid corners. The two triangles have slight curves.
You all saying the bend is the answer are retards, excuse me.
But you will get the same illusion if you made the figures yourselves witht the most precise accuracy to each other as possible.
Therefore, the bent line “theory” does not explain it.
OK, now, after reading almost all, and I say almost so that i dont insult someone i think you are ALL WRONG. The key here is not trigonometry, nor in the triangles. Let me explain. I think the answer is this – surface area and circumference are NOT in direct relation. This would mean that objects with the same circumference can have different surface area and the other way around. So – Leave the triangles aside, the difference in the two pictures is not their surface area. Its their circumference. Look closely. If you count the squares, their number is the same in both picures, its just that in the bottom picture the overall circumference of the figure is greater. Again to make i easier I will leave the triangles, they are there for confusion and go down to the numbers.
In the first picture, we have a rectangle with a surface are of 3×5 = 15
Now. In the second we have a figure that is not an exact rectagle (look only at the coloured part) but this figure stil has a surface area of 15 (just count the coloured squares). So where is the difference?As i said the difference is in the circumference. the first figure (the rectangle) has a circ. of 2×3 + 2×5 = 16 while the second has 2×8 + 2 = 18. (If you a re wondering why I add 2, I will explain. I calculated the circ. of the rectangle with sides – 2 and 8. then subtracted the non coloured bit, which was 1 (remember we are talking about sides, now squares here), but had to add 3, which is the number of coloured sides, all in all -1 + 3 therefore +2. I hope you got it. To sum up. The key in this is the question. And almost all you had it WRONG. the two coloured figures actualy have THE SAME SURFACE AREA. Question is why do two different circumference-s belong two figures with same surface areas? Now proving that in generalall cases is much tougher, but you can make it look easier with a simple exercise. Imagine a cube with a side =4. Now this has a circ = 16 and a surface area again = 16. now draw a rectagnle with a side A = 2, and B = 6. What do you notice. They have the same circ.-s = 16, but the surface area of the second is 2×6 = 12 which is NOT 16. There you go. All non linear conspiracy theories are refuted.
I got it in less than 3 minutes. I still don’t understand why everyone is talking about trigonometry and that stuff. It’s simple! It’s just a different arrangement of the shapes! Move the orange shape to the side and (a la “tetris”) watch it fall and form the hole. I would tell you more, but I don’t want to feed the beast.
It’s not as hard what everyone seems to think. There’s no “Line Bend” that creates enough area for the “hole”. The fact is, the “hole” is not part of the area for the triangle. you can’t use a trigonometric equation on the second “triangle” because it’s not a triangle. By moving the individual colored pieces, one can simply change the shape, and while it appears to be a triangle, it isn’t. The pices still have the same area, and this can be easily replicated: just make your pices [either on paint/photoshop, or with some paper] and rearrange them like on the picture. No “magic line bend” required.
The Hypotinuse is NOT a straight line in either image. it bows slightly outward in one and inward in the other. The gridlines distract your eye from noticing it.
I have not read all comments so forgive me if this is being restated. The two trinagle are not the same proportion. They have a slightly differing slope, but it is close enough to trick the eye.
The red traigle is 3/8 and the Greaan is 2/5 or 3/7.5. That .5 differsnce allows for the extra square to be created when the pieces are moved around. Thats all there is too it
I knew in about 5 min that the squares are not perfect squares in the grid.It’s the only way to explain how it’s possible to “gain” an extra space. Makes sense now as I looked down on hypotenuses they’re not parallel.
I know some couldn’t really wrap their minds around it. But it’s really on the fact that the hypotenuse is not a straight line. As was mentioned before by others.
Just a reminder (in case you’ve missed it), check this again http://www.scientificpsychic.com/mind/mind1.html
for a clear explanation.
just look at the slopes of the hypotenuse of the red and cyan triangles. they’re not the same, so it isnt a continuous straight line
Look at the orange shape and the lime shape. The orange shape has two squares on its head and three squares on its tail. The lime shape has three squares on it’s head and two squares on its tail.
On the top, the head of one shape lines up with the tail of the other. That makes a rectangle that is three squares high and five squares long.
On the bottom, the orange shape is moved so that the tails touch. The tails don’t match though. So, that creates the gap. It also changes the rectangle to two squares high and eight squares long.
The red triangle is one square higher and three squares longer than the green triangle. So, when you move the red one up, it makes up for the decreased height of the two non-triangles in the first image, like adding 1 and -1. It also stretches to cover their increased length as well.
The green triangle is only two high and therefore matches up with the second rectangle. Since it is three squared shorter than the red, it ends up stopping where the red triangle originally stopped even though it is attached to a rectangle that is three squares longer. It’s like adding 3 and -3.
There is no net change in the overall shape of the two triangles other than the gap.
I laugh at the people talking about curved lines and such. Draw it out with perfectly straight lines and you will get the same result. It’s pretty easy to figure out what the problem is as other people have pointed out. The red triangle has a base length of 8 squares and a height of 3. The green triangle has a length of 5 squares and is set 3 up from the bottom. That makes the space with the other two objects have an area of 15sq (length x width L is the base of the green;5, and W is the height of the red;3. 5×3=15). When you switch the triangles you are changing that space from 15sq to 16 sq. The length becomes 8 (base of the red triangle) and the width becomes 2 (height of the green triangle)so the area left to fill is 8×2=16sq. You can’t take two pieces that fit together and have an area of 15 and move them to complete an area of 16.
i dont see how you idiots dont see it
Sorry comment #118 (Nightmare) – but even the explanation you gave constitutes a curved line.
You can find a picture here to illustrate what you explained. http://www.scientificpsychic.com/mind/triangle3.gif
Explanation of the answer found here. http://www.scientificpsychic.com/mind/triangle1.html
@Anonymous comment 120
do you know the difference between curves and angles? Apparently not. Let me give you the definition of a curve.
curve (kʉrv)
adjective
Archaic curved
Etymology: L curvus, bent: see crown
noun
1. a line having no straight part; bend having no angular partI will admit that after going back and looking closer at the first 13×5 triangle drawn on graph paper, the 5×3 point on the hypotenuse is slightly below the grid line. So it’s an angle and not a curve. Either way it still has to do with the area of the object changing when you rearrange them. I’ll even quote from the site, “The top figure has an area of 32 square units. The bottom figure, including the empty square, has an area of 33 square units.” Which is because it’s not a triangle.
Even if you don’t get the math–it doesn’t matter. The point of optical illusions is to point out how your brain tricks you into “seeing” something a certain way, regardless of the physical reality of the actual thing you are seeing. That’s what is really the mystery here–the way our brains work!
Take a blank sheet of paper and line up the edge with hypotenuse of the red triangle. Watch as the hypotenuse of the green triangle magically jumps off of the edge of the paper!
point.
IF ANYONE OF YOU KNOWS HOW TO USE AUTOCAD, YOULL FIGURE IT OUT, JUST DRAW THE EXACT SAME FIGURE AND YOULL FIND A CURIOUS SURPRISE
The big triangle isnt a triangle!!!
i cut it out of paper. it still gives an extra square, so it is not an illusion and it is not because of bended lines. i can’t explain it, but i am really bad at geometry
This is a very interesting puzzle. I remember finding it and figuring it out when I was in 9th grade, but it took me a good while. Very intriguing. I think people are getting hung up on terminology here. There are no “bent” lines per se. All individual shapes in the top and bottom figure are identical and drawn with straight lines. The grid is not flawed either. If you add up the areas of the individual shapes (in either picture) you get 32. If you take the area of the top figure as a triangle you get 32.5 and for the bottom figure 31.5. Neither is correct, so the obvious answer, the large figures are not triangles, but then you muse find out why. The slope of the small red triangle is not identical to the slope of the small green triangle. Therefore, the total figure is not a triangle, but a quadrilateral because it’s “hypotenuse” has a slight angle change at the meeting of the red and green triangles. I don’t want to say a bend because of terminology again. All lines are straight. But the “hypotenuse” of the large figures is really two separate lines that are convex in one figure and concave in the other, accounting for the “missing” unit.
If this does not make sense to you, may I propose something. Get a piece of graph paper and draw the red (8×3) triangle and green (5×2) triangle on top of each other. To make it even more pronounced you can double or triple all distances, as that does not change the angles. Draw an 24×9 triangle on top of a 15×6 triangle using a ruler. You should very clearly see that their hypotenuses are at different angles.
Warning, here goes a rant. Now what I find amusing is the people who laugh and say how this problem is so easy, how can everyone not see it? You amuse me. Especially when you bring up, my little 13 year old mind can see this easily! Oh you. If your solution is that the shapes were moved, all I can say is, REALLY?! The rest of us didn’t notice the shapes were moved around! Thank you for enlightening us so!! Now please go back to your tinker toys and lego while the rest of us move forward with our lives by using our brains. If your answer says anything about the two triangles in the first figure producing a 5×3 cavity (15 area) while the two in the second figure they produce an 8×2 cavity (16 area) or some mention of tetris, probably followed by a comment about how you don’t believe everyone can’t see this right away, let me explain something to you. Not only are you too dense to be able to solve the problem, you are too slow to even understand that there is a problem. And that, my friends, is far worse. In the illusion, a unit of area appears to have disappeared. No matter how much you try, you cannot explain a change in area by moving things around. Its like I shuffled a deck of 52 cards and ended up with a deck of 51 cards. And your answer is, well you shuffled it. Ok. Yes I did. But that’s not the problem. Or I moved a table across the room, and it turned into a desk. Your answer is, well you moved it. That’s not the problem. There is an illusion here, and you are not even seeing the illusion, much less the explanation to it. So if you cannot even understand the problem, don’t try to act like everyone else is stupid for not seeing the answer. Understand the problem before you even try to solve it, please.
Thanks.
the two triangles shown simply arent similar triangles.
good illusion
this is not an exercise in plane geometry as much as an exercise in being able to explain the answer coherently. one person didn’t know the difference between circumference and perimeter, and another the difference between bent and curved. the illusion is that the large “hypotenuse” is a straight line, which it is not. the two small angles of the triangles aren’t equal because 8:3 and 5:2 aren’t equal ratios; therefore when you reverse the triangles the “hypotenuse” bends outward instead of inward. the increase in total volume of the large triangle is enough to accomodate the extra square. this can be proven by simply adding the four individual areas.
oops – i spelled accommodate wrong!
I don’t get it.:s
You can buy a similar problem in the form of a wooden puzzle: http://www.thinkgeek.com/geektoys/games/be62/
somthing u dont notis is the tryangles are bent diffrently but 1 thing id like to say is that i cut out the shapes out of a peice of paper and re aranged them and got the same illusions. savy? (savy means ok)
READ THIS FOR THE TRUTH
oh and 1 more thing the bend is just making people think the wrong thing (only the tryangles are bent) if u cut out your own shapes like i did u can see for yourself
If you take the smaller triangles and do a little trigonometry on them, you will see they are not similar. The smallest angle on the small triangle is 21.8 degrees, the big (red) triangle has an angle of 20.6 degrees. I tried roughly sketching this out on graph paper, and they appeared similar, but a difference of 1.2 degrees is imperceptible.
If you take the top shape and kind of fold it over on the bottom shape, so that the green triangles and the red triangles make a 2×5 rectangle and a 3×8 rectangle, respectively, then you can see this more clearly. You get a 5×13 rectangle with one piece missing, because the area of 5×13 is 65 and the areas of all the colored shapes only adds up to 64.
There is no bending, just dissimilar triangles.
The “whole” triangle is not REAL triangle, it’s just a look-alike.
The red&green triangles appear to be similiar (to have equal angles), but actually their angles differ. If you assume all the sizes of the shapes (which form the whole picture) are determined exactly by the number of squares they occupy, you can easily calculate the angles of the red&green triangles, using trigonometry.
This is a mathematical proof that the hypotenuse of the top (whole) triangle is not actually a straight line, which means that this object is actually not a triangle, which means that its area cannot be calculated (accurately) with the “right-triangle-formula”, but rather “the-sum-of-its-parts” approach should be used.
The conundrum here is that, the point where the red&green triangles meet appears to be exactly on top of the grid, which gives the erroneous impression that their sides are well-defined. In reallity, this picture is ambiguous. If the grid represents the sizes accurately, then the sides of the red&green triangles can’t be measured with whole numbers (integers), but irrational numbers should be used instead.
So… my conclusion is that:
1) The “whole” triangle is not REALLY triangle (its hypotenuse isn’t straight line)
2) If you get a grid of squares and draw a STRAIGHT line in such a way that it forms the hypotenuse of a right triangle with sides 5 and 13, you’ll notice that this line doesn’t go through any of the grid intersection-points (it will come close, but not exactly over where the grid lines intersect)
3)This is one EXCELLENT visual illusion.
[i]Anonymous says:
February 23, 2009 at 10:36 pm
OK, now, after reading almost all, and I say almost so that i dont insult someone i think you are ALL WRONG. The key here is not trigonometry, nor in the triangles. Let me explain. I think the answer is this – surface area and circumference are NOT in direct relation. This would mean that objects with the same circumference can have different surface area and the other way around. So – Leave the triangles aside, the difference in the two pictures is not their surface area. Its their circumference. Look closely. If you count the squares, their number is the same in both picures, its just that in the bottom picture the overall circumference of the figure is greater. Again to make i easier I will leave the triangles, they are there for confusion and go down to the numbers.
In the first picture, we have a rectangle with a surface are of 3×5 = 15
Now. In the second we have a figure that is not an exact rectagle (look only at the coloured part) but this figure stil has a surface area of 15 (just count the coloured squares). So where is the difference?As i said the difference is in the circumference. the first figure (the rectangle) has a circ. of 2×3 + 2×5 = 16 while the second has 2×8 + 2 = 18. (If you a re wondering why I add 2, I will explain. I calculated the circ. of the rectangle with sides – 2 and 8. then subtracted the non coloured bit, which was 1 (remember we are talking about sides, now squares here), but had to add 3, which is the number of coloured sides, all in all -1 + 3 therefore +2. I hope you got it. To sum up. The key in this is the question. And almost all you had it WRONG. the two coloured figures actualy have THE SAME SURFACE AREA. Question is why do two different circumference-s belong two figures with same surface areas? Now proving that in generalall cases is much tougher, but you can make it look easier with a simple exercise. Imagine a cube with a side =4. Now this has a circ = 16 and a surface area again = 16. now draw a rectagnle with a side A = 2, and B = 6. What do you notice. They have the same circ.-s = 16, but the surface area of the second is 2×6 = 12 which is NOT 16. There you go. All non linear conspiracy theories are refuted.[/i]
So, dude… if you can’t figure out why the apparently straight line isn’t really straight, you can pick your favorite ruler and see this for yourself. Or maybe they’re not that straight anymore? It’s a fact, not a theory. With trigonometry, you can (on theory) see exactly which portions of the elements form the missing square.
long story short, anyone smart enough to recognise the problem already knows the answer. two of the same shapes cant have a different area… so whatever cant be true, isn’t. the shapes are either different, or the area is the same. in this case the shapes are different. END OF STORY.
but ill keep talking anyways
if simple theory doesnt work on you you can do as one person did somewhere above and prove it with simple geometry. he figured it out by calculating all dimensions looking for the inconsistency, and found it.. a different angle on the two triangle.
you cannot argue that you built the pieces and it still works, because that is 100% correct, you will accomplish the exact same effect. the illusion here isnt how it is drawn, its just taking a small area, hiding it over a long distance or focusing it in a small space, the small angle is just real hard to see with the eye, and will be only hidden better by trying to tape little pieces of cut out paper together. now if you cut these out v ery accurately, and large enough to work with, and very accurately traces the perimiter, rearanged them as shown, you would see a gap totaling the same area develop, spread out along what is a bent line. basically a waste of time thought, all you did was copy what was already shown on your screen.
some of you are making this way way too complicated, others are too stupid to understand the problem. the trick to this isnt math, geometry, trig, its simple theory. the area an object cannot dissapear by rearranging pieces of it, but the resulting shape can change. so the shape of the two triangles MUST be different. and it is.
the missing area is hidden along the length of the hypotenuse by the different angle created.
This is quite simple to figure out. The slope of the hypotenuse of the two triangles are not equal. The smaller green triangle’s hypotenuse has a slope of 2/5 while the larger red triangle has a slope of 3/8. Since these slopes are not equal, then the hypotenuse of the assembled triangle is not a straight line.
In the top figure the long hypotenuse is actually concave (bowing inward) while in the bottom figure the hypotenuse is convex (bowing outward) making the contained area of the lower triangle greater than that of the top.
The guy talking about circumfrences is a retard. the line IS NOT STRAIGHT! the red triangle and the green triangle have DIFFERENT ANGLES. This makes for the big shape being created to NOT be a triangle, it only appears to be a triangle.
The longest line in both shapes are not straight, they are bent. The ankle of the red shape is not the same as the green one. When we want to know the size of the shape we draw a rectangle and calculate its surface area (of the first shape). That is 13*5=65. Then we calculate the first shape without dividing 65 by 2. First we calculate the green triangle. 5*2\2=10\2=5. The red triangle is 8*3\2=24\2=12. The green shape + the red shape= 5+12=17. Next we calculate the rectangle made out of the orange and green shapes. That equals 5*3=15. 17+15=32. So 65 – 32=33. That means the first shape’s size is 32 and the second shape’s size is 33. That extra one in the second shape gave it enough room to put one extra square.
Yes most of the comments are right.
I actually calculated the area that is created by the bent hypotenuse. The first picture has 0.5 area less than a straight hypotenuse would have had. The second picture has 0.5 more. add those two up. getting 1 whole box
oh, i see it now
the spare square comes from the orange part cuz it doesn’t fit horizontally to the green part.
and also the first ‘triangle-like’ shape is not a triangle
this gets into the 2/3 .. 23 .. .666 deceptive truth.
The second triangle is greater than the first few millimeters, 1 millimeter to the left of 1 millimeter and one millimeter from the top right, and because of this that space remains empty, is tested by me, porblema solved!
i dont get it
WEIRD
★★★★★★★★★★★
1. The hypotenuse is never straight. Some of you that say it is, can try with a piece of paper and find out it’s impossible, because a tiny bit of the orange piece will stick out of the hyp. anyhow.
2. Just grab a ruler or anything perfectly straight, place it on the hypotenuse on your monitor. (They are never straight. The thin slice of the area missing (in the first triangle) or in excess(in the second), answers for where the empty area went.
Do the maths on your own, but i do believe we need no numbers and calculations for this.
That’s a pretty awesome puzzle. We immediately assume it’s a right triangle, when it actually isn’t; very tricky.
Hi,
I didn’t read all the comments but I saw that some of you have found the answer. Some of you tried to cut the figures in a paper and said that this is true. Well, they forgot one IMPORTANT thing: they forgot to put it in a canvas, then they should have seen that by changing the position of the figures, it didn’t fit anymore in the canvas. The extra is very small, it corresponds to 1 square distributed equally around the whole figure.
Now to the scientific explanation: il you move some figures in a different way, it can’t gain or loss size (English is not my mother tongue so I don’t know if I’m clear enough), those figures will always have the same size, unless it is ice figure : ) But we you keep saying that we see a square in the example. Well then that means that the figures in our case have shrinked to fit the canvas. That’s why people who tried this in a drawing software like photoshop realise that the seconf figure can not fit in the same canvas.
Sorry, nothing grows or shrinks when you move then, they always keep the same size.
so how is this ment ot be an illusion? its just a bunch of shapes…
easy .. observe well longest line is bended..! THATS ALL in this illution..
The area of the second triangle is bigger.
spoiler:
the first assumption that most people make is that all of the parts put together as they are make up a triangle, this is not true.
the smaller green triangle is 2 x 5 (40% slope)
the larger red triangle is shallower at 3 x 8 (37.5% slope)
on the top set, the green and red triangles create a concave(inward sloping) surface, and when switched they create a convex(outward sloping) surface giving you the size difference.
MuntainDW is right , this curvature that he spoted does not be so clear due to the thick black lines that are used .You can see that litle diference if you count from left to right five boxes and then two up , if you compere the up (red) and the down (green) you will see tha the coloured aera is different , before and after the critical dot .
The key of the illusion is that the diagonal lines are not straight.
Is not a triangle, 2:5 is not 3:8, wrong, check you, the hole always exist, only need paint a triangle, impossible partition of parts with this.
thankyouthankyouthankyouthankyouthankyou, you just saved me from 3 years of puzzling this
The trick is that the borders (black lines) in the second picture are thicker in a nuber of places.
it should be yellow. it’s just confusing because the colors are changed.
This is nothing but a scam, it’s full bullshit, trie to draw it with 3 straight outer lines and it’s impossible to get the same area of the triangles. The angle of the slopes differ so little that you almost can’t see it on the drawing. It’s maybe even less than 0.05 degrees!
But the trick is quite cool :)
Its from the “Red” triangle.. have a look close to the Red triangle in first and second picture, will get the answer.
I’ve finally figured it out….
A bit difficult to explain…
The “issue” is that with the large red triangle (top picture), the drawing seems to put each of its corners into the corners of a square, in the graph.
But one of the corners actually does not go into one of the corners of a square (the fact of which is disguised, by using the black thick line).
And so the trick is… when a new picture is drawn, it is drawn in a way, where the red triangle is a full triangle, with all corners fitting exactly into the corners of the squares – this is not representative of the triangle which is in the top picture. The diagonal line should be shorter)
If you print and cut out the shapes on paper, things might become clearer.
http://www.break.com/index/the-impossible-puzzle-1936440
explain this? its used with real physical pieces.
the diagonal side is taller in the second triangle
It’s quite simple, actually.
Green triangle: 5×2
Red triangle: 8×3
When you move the red on top, and the green to the bottom, you get 1 extra square of height and 3 squares less length.
To account for the height and length difference, you move one of the square~ish blocks down, giving you 1 sqr less in height, and 3 sqr extra in length.
Now, these blocks aren’t a perfect fit in this position, due to the ‘neck’ being 3 sqr and the base being 2.
Thus, you get that empty square in the bottom.
While the total triangle keep the same shape.
true
Agreed :)
you are wrong. these figures are not a triangle. they seems to be a triangle but in fact they are 2 different quadrangle. (this is an illusion, remember?)
very difficult.