Finally, An answer!
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Senior Member
Joined: Jul 2007
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From: Columbus, OH
Finally, An answer!
This has stumped me for years. It's about time I found the solution:
We've all heard that you can lie with statistics. That by fudging the numbers just a little bit, or by asking the wrong question, making the graphs look bigger/smaller than they should be, and so on, you can exaggerate or completely fabricate the truth. And we all know that there's a fundamental problem with most discussion on the internet because you can't prove anything anyone says, so it all boils down to a matter of opinion, shouting, and logic puzzles designed to get the other person to concede "yeah, maybe." Well, never has this been more obvious to me than in the Triangle Problem. I don't know how old this is, or where it comes from, but I've seen it in a handful of places on the internet, even in another language (maybe that's where it originated?). On first glance, it looks impossible, but after doing some research, I saw the subtle lie that was being perpetuated right before my eyes. I give it to you exactly as I found it (well, except for adding my website to the corner so I get at least a little promotion from hot linkers).
Try and figure it out before continuing on. Some people intuitively "get" it, but you have to understand what's going on. Other people don't get it, even once it's been explained to them. This is obviously impossible. If the triangles are the same size, then how can we create an area that has 1 extra space? And it's obvious that the triangles are the same size - 8 long by 3 high, and 5 long by 2 high. Staring at the picture for a while you begin to notice subtle things that seem wrong but can't possibly explain such a large hole. If you look at the bottom picture you'll see that the green triangle is exactly 2 high, but the if you look at the top drawing, the red triangle passes just under that spot in the same place. And if you look at the top picture, you'll see that the red triangle is exactly 3 high, but the picture on the bottom crosses just over that spot in the same place. The top triangle just looks a sleight bit bigger than the bottom triangle, but by looking at the red & green triangles themselves, they're always the same number of boxes, and more importantly, the triangle as a whole is always the same size: 13 x 5. The truth is, this isn't a triangle you're looking at. It's two different four sided figures. The apparently straight line that connects the bottom left to the top right isn't straight. It's two different lines, and that accounts for why the top drawing doesn't line up with the bottom one exactly, even if the shapes appear to be the same basic size. Let's forget about the smaller triangles for a moment and draw the boxes. One is 3 x 5, and the other is 2 x 8. (my apologies to anyone who may be color blind, I can re-do these drawings if you need it).
If you look closely, you'll see that the 3 x 5 red box is on the inside of the gray triangle area, and the 2 x 8 green box is on the outside. Let's zoom in a bit so you can see that easier. 
That's such a small difference though, it can't account for an entire box being created, can it? Let's look at the two triangle again, this time, superimposed one on top of the other. Here you can see clearly that the 2 x 8 green box is just a little bit outside the line, and the 3 x 5 red box is just a little bit inside the line. Neither falls exactly on the line. You can also see more clearly the difference between triangles created by the green and red boxes. 
and again, zoomed in 
Okay, so some of you are sayi
We've all heard that you can lie with statistics. That by fudging the numbers just a little bit, or by asking the wrong question, making the graphs look bigger/smaller than they should be, and so on, you can exaggerate or completely fabricate the truth. And we all know that there's a fundamental problem with most discussion on the internet because you can't prove anything anyone says, so it all boils down to a matter of opinion, shouting, and logic puzzles designed to get the other person to concede "yeah, maybe." Well, never has this been more obvious to me than in the Triangle Problem. I don't know how old this is, or where it comes from, but I've seen it in a handful of places on the internet, even in another language (maybe that's where it originated?). On first glance, it looks impossible, but after doing some research, I saw the subtle lie that was being perpetuated right before my eyes. I give it to you exactly as I found it (well, except for adding my website to the corner so I get at least a little promotion from hot linkers).
Try and figure it out before continuing on. Some people intuitively "get" it, but you have to understand what's going on. Other people don't get it, even once it's been explained to them. This is obviously impossible. If the triangles are the same size, then how can we create an area that has 1 extra space? And it's obvious that the triangles are the same size - 8 long by 3 high, and 5 long by 2 high. Staring at the picture for a while you begin to notice subtle things that seem wrong but can't possibly explain such a large hole. If you look at the bottom picture you'll see that the green triangle is exactly 2 high, but the if you look at the top drawing, the red triangle passes just under that spot in the same place. And if you look at the top picture, you'll see that the red triangle is exactly 3 high, but the picture on the bottom crosses just over that spot in the same place. The top triangle just looks a sleight bit bigger than the bottom triangle, but by looking at the red & green triangles themselves, they're always the same number of boxes, and more importantly, the triangle as a whole is always the same size: 13 x 5. The truth is, this isn't a triangle you're looking at. It's two different four sided figures. The apparently straight line that connects the bottom left to the top right isn't straight. It's two different lines, and that accounts for why the top drawing doesn't line up with the bottom one exactly, even if the shapes appear to be the same basic size. Let's forget about the smaller triangles for a moment and draw the boxes. One is 3 x 5, and the other is 2 x 8. (my apologies to anyone who may be color blind, I can re-do these drawings if you need it). If you look closely, you'll see that the 3 x 5 red box is on the inside of the gray triangle area, and the 2 x 8 green box is on the outside. Let's zoom in a bit so you can see that easier. That's such a small difference though, it can't account for an entire box being created, can it? Let's look at the two triangle again, this time, superimposed one on top of the other. Here you can see clearly that the 2 x 8 green box is just a little bit outside the line, and the 3 x 5 red box is just a little bit inside the line. Neither falls exactly on the line. You can also see more clearly the difference between triangles created by the green and red boxes. and again, zoomed in Okay, so some of you are sayi
(kidding)
Joined: Aug 2007
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RE: Finally, An answer!
Let's see....Sin this, Tangent that, Arctan what?..... it takes a 178.75 Degree angle on the hypotenuse (which is really two intersecting lines) for the bottom one and a 181.25 Degree angle on the top one for this to work. Because the outside angles only differ by 2.5 degrees, the lines would seperate at 1.25 degrees, which is negligible for most applications. The trick is that the intersection of the two lines that make up the hypotenuse are not in the same spot but rather 38% from the top or bottom, depending on the picture. The lines of the two "hypotenuse" thus form their own quadrilateral, which happens to be a symetric parallelogram. The eyes are fooled because the angle of each of the lines with respect toan arbitraryplaneis exactly the same. The straight hypotenuse would have angles of68.96 and 21.04 degrees. By manipulating it, the first triangle has angles of 68.185 and 20.565, and the second triangle has angles of 69.435 and 21.815. We can then deduce that each of the lines, when superimposed, is parallel to the opposing line and would not be recognizable as having migrated about .085 units up the graph.
Also, the red draws the eye away from the sharper angle in each picture, lol. And the human eye is not trained to pick up on a 3% difference in the overall size of the triangle regardless of whether the box is removed or not.
Some of the explanation pictures are not actually correct. The superimposed lines should be parallel through blocks 6,7,8 and come back together at the angles listed above. The 4th and 5th picture in the post with the red and green lines are the correct solutions.
Good post. I love puzzles.
Also, the red draws the eye away from the sharper angle in each picture, lol. And the human eye is not trained to pick up on a 3% difference in the overall size of the triangle regardless of whether the box is removed or not.
Some of the explanation pictures are not actually correct. The superimposed lines should be parallel through blocks 6,7,8 and come back together at the angles listed above. The 4th and 5th picture in the post with the red and green lines are the correct solutions.
Good post. I love puzzles.
Banned
Joined: Dec 2005
Posts: 3,837
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From: Indian Harbor Beach, Florid
RE: Finally, An answer!
this is a mistake in assumptions and/or optical illusion not really a statistical trick or misstatement.
A statistical trick is something like '55% of people are pro-choice' and it is restated as 'most americans support killing children'
A statistical trick is something like '55% of people are pro-choice' and it is restated as 'most americans support killing children'
Senior Member
Joined: Feb 2007
Posts: 571
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RE: Finally, An answer!
ORIGINAL: Ty
Let's see....Sin this, Tangent that, Arctan what?..... it takes a 178.75 Degree angle on the hypotenuse (which is really two intersecting lines) for the bottom one and a 181.25 Degree angle on the top one for this to work. Because the outside angles only differ by 2.5 degrees, the lines would seperate at 1.25 degrees, which is negligible for most applications. The trick is that the intersection of the two lines that make up the hypotenuse are not in the same spot but rather 38% from the top or bottom, depending on the picture. The lines of the two "hypotenuse" thus form their own quadrilateral, which happens to be a symetric parallelogram. The eyes are fooled because the angle of each of the lines with respect toan arbitraryplaneis exactly the same. The straight hypotenuse would have angles of68.96 and 21.04 degrees. By manipulating it, the first triangle has angles of 68.185 and 20.565, and the second triangle has angles of 69.435 and 21.815. We can then deduce that each of the lines, when superimposed, is parallel to the opposing line and would not be recognizable as having migrated about .085 units up the graph.
Also, the red draws the eye away from the sharper angle in each picture, lol. And the human eye is not trained to pick up on a 3% difference in the overall size of the triangle regardless of whether the box is removed or not.
Some of the explanation pictures are not actually correct. The superimposed lines should be parallel through blocks 6,7,8 and come back together at the angles listed above. The 4th and 5th picture in the post with the red and green lines are the correct solutions.
Good post. I love puzzles.
Let's see....Sin this, Tangent that, Arctan what?..... it takes a 178.75 Degree angle on the hypotenuse (which is really two intersecting lines) for the bottom one and a 181.25 Degree angle on the top one for this to work. Because the outside angles only differ by 2.5 degrees, the lines would seperate at 1.25 degrees, which is negligible for most applications. The trick is that the intersection of the two lines that make up the hypotenuse are not in the same spot but rather 38% from the top or bottom, depending on the picture. The lines of the two "hypotenuse" thus form their own quadrilateral, which happens to be a symetric parallelogram. The eyes are fooled because the angle of each of the lines with respect toan arbitraryplaneis exactly the same. The straight hypotenuse would have angles of68.96 and 21.04 degrees. By manipulating it, the first triangle has angles of 68.185 and 20.565, and the second triangle has angles of 69.435 and 21.815. We can then deduce that each of the lines, when superimposed, is parallel to the opposing line and would not be recognizable as having migrated about .085 units up the graph.
Also, the red draws the eye away from the sharper angle in each picture, lol. And the human eye is not trained to pick up on a 3% difference in the overall size of the triangle regardless of whether the box is removed or not.
Some of the explanation pictures are not actually correct. The superimposed lines should be parallel through blocks 6,7,8 and come back together at the angles listed above. The 4th and 5th picture in the post with the red and green lines are the correct solutions.
Good post. I love puzzles.
you can call me captain thread stealer :-)
Joined: Aug 2007
Posts: 251
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From:
RE: Finally, An answer!
Captain Thread Stealer -
It was. I recently sold it, but it still holds a place dear to my heart. That thing was a monster. It would take 20+ lbs of boost and not even flinch. It was my daily drivenproject for about 5 years, during which time I think I upgraded every single part on the car, lol. I, too, have a love for turbo rotary power. There is nothing like hearing your wastegate scream in pain at 9000 RPM.
It was. I recently sold it, but it still holds a place dear to my heart. That thing was a monster. It would take 20+ lbs of boost and not even flinch. It was my daily drivenproject for about 5 years, during which time I think I upgraded every single part on the car, lol. I, too, have a love for turbo rotary power. There is nothing like hearing your wastegate scream in pain at 9000 RPM.
