OK I have looked, and looked.. counted squares I still can't figure this out...

I dont understand what answer should come. I've looked and looked also, and cant see what answer i should be finding.

here is a hint...go into photoshop and overlay the triangles ontop of each other while using some transparency....

I did that in photoshop. and overlayed them. all that seems to do for me is confirm that all the pieces are exactly the same. i dont see what "the answer" is. and i dont understand how the area of a triangle can be reduced while all its lengths stay equal. SO CONFUSING!!!!!! :)

*** SPOILER ALERT ***

It's easy...

The question is (if you don't understand the question...) Why is there an empty space, if the four small pieces are all the exact same size (when comparing the top to the bottom).

The answer is that the two diagrams don't take up the same amount of space elsewhere. It LOOKS like the top 4 pieces make up a triangle, when in fact it's really a 4-sided figure. The diagonal "line" runnning along the top left (of the first diagram) is actually two line segments with a small inward bend. Two straight lines, but meeting at a wide angle (not quit 180 degrees, which would be a straight line.. more like 179 degrees (estimated))

In the second diagram, it's a small "outward bend" along that same edge. (it's two straight lines again, but they do not run at the same slant relative to the page, so they meet at an angle.)

The difference between that inward bend and the outward bend, when spread across that line, actually amounts to one whole square. (the math is easy on this, trust me)

To further prove it, look at the point of inflection on the top diagram (where the red triangle touches the green triangle.) In the top diagram, it's exactly on the interersection of lines on the grid. Now look at that same (X,Y) point on the lower diagram, which is now within the Red triangle. See how far it's within the Red?

In looking at the top diagram, your eye tends to assume that it's one giant triangle made up of 4 pieces. But hold a piece of paper up, with the edge along that diagonal line, and you'll see that it's really not one straight line after all!

Thank you! Stupidly enough i saw that bend in photoshop. But just thought it was a tiny mistake. Not enough to explain anything. How wrong i was :).
Now i can sleep again

There is actualy an analitical way to find out what's wrong.

The area of the "triangles" should be the sum of the areas of the objects composing them.

Now, this sum is:

2*5/2 = 5, for the green triangle
3*8/2 = 12, for the red
3*5 = 15 for the two L shaped objects.

TOTAL AREA = 32

The area of the big suposed triangle, however, is 5*13/2 = 32,5

The only explanation is that the big object it is not a triangle.

Other way would be to calculate the sin of the sharp angles for the green and red triangles. It's not the same and this tells us there are two segments and not one straight line.

Thank you

STILL DONT SEE IT

hint:


look at the two nontriangles...the answer lies with them

hah thank you. This problem was extra credit in my plain & solid geometry class, so thanks a ton =]

Oh poop i alerjik 2 geomutry .But it still makes perfect sense and i would have fuigured it out if i wasnt so lazy

my brother got so mad i was stairing at it so long!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

oh.......i get it now

BALLS

I am ashamed to admit that it took me as long as 2 minutes to figure this out. The answer is so obvious and it will only take one VERY short paragraph to explain it. Comment # 10 is correct.

thanks for this illusion

o I got this one right away, its the transfunction of the x axis and the velocity of the y axis when put in a parallel conversion equilevent to the triangles centrifical force.

Im just kidding ya, i was just trying to act smart

la réponse est que la grille n'est pas régulière elle rétrecit de haut en bas.

Graphs with colored shapes = How it was made-Photoshop. PP is probably responsible for all the impossible objects.

thank you. two supposed geniuses couldnt get this ....u.c. systems....lol

wow this only took me about 40 seconds to figure out! since the red and cyan triangles are different sizes the red one will ofcourse have to stretch out the two ther peices.....
did that make sense???

wow this only took me about 40 seconds to figure out! since the red and cyan triangles are different sizes the red one will ofcourse have to stretch out the two ther peices.....
did that make sense??? oo if u like adventurequest then go here for a game that youll love

I Never thought of that, Thanks a LOT!

I Have Been working on this homework for hours.
But now I see it
thanks

its not bent....i made my own...very carefuilly so its straight....cut out the shapes....not bent....dumbasses

i dont understand al this technical talk I still don't get it!!!

Where can I find an answer to this?


cheers

Ok for all the people who STILL don't get it, here's a simple explenation:

The longest side of the triangle is called the hypotenuse, but lets call it h. Ok, if h ISN'T straight, then it must be in a curve, right?
Right. So, if it's curved, then it's either adding more space if it's curved outward, or removing space if its curved inward.
In the first diagram, h is curved inward, so there is some triangle missing. In other words, its smaller than it looks.
In the second diagram, h is curved OUTWARD, so there is MORE triangle than it seems. In fact, there is exactly 1 sq. unit's worth of square added.
So, it's really simple. If you STILL confused, email me at vgm10.000@hotmail.com, and i'll try to get some simple geometry into you!

wow me and my friends have never figured this out for like a year..... pretty pathetic ( but i didnt have photoshop euther) now im gonna tell my homies

ty all for letting me understand this "simple" equation and i wonder who made this?...

well what happens if you make the line straight then do it?

i made one by paper... it still makes a space...wuddabou'dat?

i don't get it all!it's really funky,so keep it up!!!!

****HEAD ACHE ALERT**** WOAH

42

The blue triangle on top makes the rectangle 3X5 (15 squares)
The red triangle on top makes the rectangle 2X8 (16 squares)
keep it simple stupid

what's the problem?
simple math'...
the light green + the red = 13 squers

the light green + the orang + the darker green = 12=]

it's really easy.......

its not so simple like everyone says it is but i will try to explain it.

going back to the line not being straight, each seperate figure is made up of straight lines, however, the slopes of the two triangles are different. so lets just throw in some random number like the angle of the red one is 40 degrees and the angle of the green is 45 degrees. they now in the first one form a line going inward of the triangle. now take them and put them like they are in the second "triange" they form an outward line. this gives it the extra space to make that extra square.
this is really explaining number 5 in a more sensible way.
also try showing it to a math teacher, they might be able to explain it better.

it has to do with the slopes of the different triangles (red and green). then after figuring out the slopes its along the lines of #5

it has to do with the slopes of the two triangles (red and green) then along the lines of #5

don't listen to these idiot jokers who think they are being cute by giving long explanations about how it's the curve of the lines

it has nothing to do with the curve of the lines, you can take the top image into photoshop and rearrange the pieces and you'll still come up with a space

lol guys its like this
_
convex / (larger)

concave _/ (smaller)

the triangles have different slopes, if you switch their places it'll change the entire triangle from concave to convex, or vise versa

I have constructed the shapes in Freehand, a with exact numeric values, no curve lines, and I still get the effect.
I really don't think that the answer is the one of the curved lines

here is the solution :
http://www.helen.co.il/puzzle/puzzle.pdf

alright all this dumb talk about curved lines and such is plain stupid and too boring for me to read so if its right then whatever but i suspect its not so let me put in my speculations.

From what i see obviously the answer is :the red triangle has been moved on top and the green one on bottom right? right. problem is the green one wouldnt be able to stretch out far enough to cover the same area as the red one does since its smaller but the answer to that problem is the orange non-triangle (dont know what you call it) which when placed where it is on the 2nd picture stretches out the green triangle far out enough to cover the same area. but in stretching itself out it leaves a little friggin square that started this dumb debate with even dumber people. it took me 5 minutes to figure this out, 3 of which were spent reading the other comments

Ruben's answer link is correct. If all of you morons who still think it has nothing to do with curved lines need prrof, check it out. You cannot make a perfect triangle and cut out the pieces and rearrange into another identical size triangle with a gap. Those of you who say you used paper, freehand, your mother's pantyhose, are wrong. The triangles you are making are not identical.

ok, I drew out the figures on graph paper using the exact square by square measurements. I cut them out and put them side by side yet they still seem to occupy the same amount of space. Draw the geometric figures on a sheet of graph paper useing a ruler and the exact measurements in the illusion. It still works even with a consistantly straight line. WTF!?!?!?

I figured it out- and I'm only 8- but I can't explain it..... SORRY!

Ok, this is the final answer whether the lines are straight or curves:

if they where straight, the proportion would have to be:

2:5 = 3:8
dark green red
=>2 = (3:8)*5
=>2 = 15:8

BUT: 15:8 is not 2...
sooooo:
the haven't the same proportion.
=> the left angle of the dark green triangle is bigger than the left angle of the red one.
=>the first BIG triangle is concave, the second is convex.

q.e.d.

Ok, the so called authorities posted here have not yet impressed me with a plausible explaination.
Is the illusion that the diagonal line isnt straight when it looks straight?
I dont think so. and the difference makes another squaare? no , i think that is too easy. that would not follow. if there were more area in one than the other than it would be just that, one has more area because of the curved line.
but i think it is more interesting than that.
It appears that the two constructed trangles have the same amount of units x and y, assuming they are true triangles and that would mean they have the same area, However within the same area, the same peices reconfigured fit the same area as the larger constructed triangle. so the illusion is that i have a grasp of its reality.
I printed it out
cut out the top peices
and they act just as you would expect
they fit over the bottom and make a space
but it the sum of the area of all the peices = the area of the larger construct, then where dows that extra space get suptracted from the sum of the little peices , just by being configured within the same area?

even if there was no curve in the top diagonals
i still think it would be the same result
as yet i am not impressed by any explaination or fancy sources of official looking diagrams.
i am still shaking my head over this.
i think essentially , that even if the so called curved lines were truely straight, the effect would still be the same , so , that would blow the curved line theory out of the water.
so the curved line dosnt explain where the blank square came from.
Unless im really missing something....

Some how it seems that the sum of the area of all the little peices does not necessarily = the total area of the larger constructed triangle shape, Why?!?
I dont know!
yeah the little square poped up and the same peices fit in the same area of the larger triangle construct.
but if all the lines were true straight lines
it still wouldnt explain the space,
the real explaination i think is far mor brainier profesorial than any of us so far...

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