The top triangle has a slightly concave slop. While the bottom triangle appears to have a slightly convex slop. Therefore, the area of the top triangle is going to be less than area of the bottom. The difference in the areas, here described, would render the white space in the bottom triangle. The triangles are not true right triangle. Actually, they are not Euclidean triangles at all.

Well, the different slopes of the two small triangles comprising the bits and pieces "triangles" would account for the curved slopes of the "triangles."

We have to ignore the outer triangle, throw it away completely.

(5*3=15) != (8*2=16)? As in the area of the "rectangles" made up of the yellow and green shapes.

One "rectangle" is 3X5, the other is 2X8. The difference is the blank square.

The problem is that the split of the rectangles into two shapes must follow integer factors of the total areas in order for the inner shapes to be interchangeable.

If you split the 5 into 2 + 3, you have to split the 13 into 5.2 + 7.8, not the 5 and 8 shown.

When you re-arrange the sections from the rectangle, the top layer would be 5.2 + 3 = 8.2 (so the red triangle would not line up properly), and adding the green triangle at the end would make the total width of the shape 13.4, not the 13 shown.

Well, there is a “Curry’s Paradox” indeed … but nothing to do with Paul Curry the magician.

See … http://plato.stanford.edu/entries/curry-paradox/

The geometrical dissection trick attributed to Paul Curry is not a paradox at all (though referred as such in many, many places … even the Stanford page above incorrectly calls it the “Geometrical Curry Paradox” … then again, they were just eliminating it as a possible source of confusion).

Here, it is appropriately called for what it is … a dissection fallacy:

1- the dissection fallacy and other such examples apply to two different shapes arrived at with the same component shapes. Thats is not the case here: the two outside shapes are the same, apart from the missing square.

2- Like Tom says, there is no trickery: cut out the shapes and try it. There is no slight of hand or slight skewing or anything like that.

Just a consequence of using integer arithmetic to define areas that are not integer. A "rounding" error, if you prefer.

Your point (1) is wrong - the two outside shapes are not the same, they only look the same because of a deliberately inaccurate presentation.

You point(2) is misleading - you don't need to cut them out, you only need to draw the figure accurately and you will see that the original is a lie.

Your last paragraph contradicts your point (2) - "using integer arithmetic" is exactly the "slight skewing" - one is the logical, the other the physical explanation of how the eye is being deceived.

This is, of course, a good analogy to the voodoo tuner who tells you that "I saw X, I did Y, everything went faster - trust me Y is a good thing". If X and Y are badly observed it's easy for the conclusion to be false.

Think about my demonstration of the /*+go_faster*/ hint - it's the same thing. I can SHOW you cases where putting this (pseudo-) hint into the SQL makes the query run faster - really - and if I don't describe accurately what I'm starting with you have to believe that the hint really works.

and now that I'm home, had dinner and coffee, I can perfectly see your point. Indeed, it's a deception.

They are likely taking advantage of the thickness of lines and the fact that eyesight fails to measure them properly when put next to each other.

The difference is very small, there is a hint of it in the hypotenuse of the large triangles and how it intersects the grid along its length. The result would be very close to the 5.2 and 7.8 you pointed out.

But I'm still trying to explain to myself how the yellow and green shapes can ever make a rectangle anywhere near 2x8. Which it most likely isn't anyway. By the time I've finished the Port I'm sure I'll have it figured out!

Suddenly this has started to appear all over the internet. I got atleast 4 email forwards ofcourse getting me stumped. Jonathan's explanation sets my confused mind to rest.

Okay, this is stupid. The triangles are not similar to each other. One is 2x5 and the other is 3x8. If you make the triangles congruent by changing 3x8 to 3x7.5, you have the same slope. Now, with this new adjustment, you have the second rectangular portion being 2x7.5 (15) which has the same area as 3x5 (15.. duh). So there you go. It's a stupid slope trick. Of course the stupid triangles are gonna be diferent if the slopes aren't the same. hmm.. that took like 5 minutes to figure out. What a stupid "paradox." It's just an eye trick.

As for the reason why the blank space is there 3x5 and 2x8 are 15 and 16. That space is supposed to be filled in. But they took it out for kicks so they could make a "paradox." Again.. STUPID STUPID STUPID.

that you looked at it for five minutes or the "trick"? :) That space is not supposed to be "filled in", it doesn't work if it were.

Yes it is the slope. Yes it is a trick of the eye.

Doesn't make it stupid - you cannot see the forest for the trees. It happens all of the time - we expected something and our eyes let us see what we expected.

The views expressed are my own and not necessarily those of Oracle and its affiliates. The views and opinions expressed by visitors to this blog are theirs and do not necessarily reflect mine.
I've been using Oracle since 1988. I've been working at Oracle since 1993 (version 7.0). I spend way too much time working on asktom.oracle.com...

## 17 Comments:

The top triangle has a slightly concave slop. While the bottom triangle appears to have a slightly convex slop. Therefore, the area of the top triangle is going to be less than area of the bottom. The difference in the areas, here described, would render the white space in the bottom triangle. The triangles are not true right triangle. Actually, they are not Euclidean triangles at all.

the triangleS are right - the triangle assembled from the bits of stuff is not a triangle :)

The slopes of the two small triangles are different.

Or, if my straight edge and eyes deceive me, the space difference could be adjustments in the border widths of the constituent shapes.

But, honestly, the slopes do not appear to be straight lines.

Well, the different slopes of the two small triangles comprising the bits and pieces "triangles" would account for the curved slopes of the "triangles."

Nice puzzle again Tom.

You've had one of my favorite puzzles on your desk for almost a year now... any luck yet? ;)

We have to ignore the outer triangle, throw it away completely.

(5*3=15) != (8*2=16)?

As in the area of the "rectangles" made up of the yellow and green shapes.

One "rectangle" is 3X5, the other is 2X8. The difference is the blank square.

The problem is that the split of the rectangles into two shapes must follow integer factors of the total areas in order for the inner shapes to be interchangeable.

It's not a puzzle, it's a piece of deception.

The initial triangle is 5 by 13.

If you split the 5 into 2 + 3, you have to split the 13 into 5.2 + 7.8, not the 5 and 8 shown.

When you re-arrange the sections from the rectangle, the top layer would be 5.2 + 3 = 8.2 (so the red triangle would not line up properly), and adding the green triangle at the end would make the total width of the shape 13.4, not the 13 shown.

Apparently, this is known as Curry's Paradox. It's well-documented here:

http://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml

Also, if you're a math geek and/or puzzle maven, don't go here:

http://www.cut-the-knot.org/ unless you want to waste many hours....

Lot's of classics there....The Nim game, the Monty Hall Problem, etc, etc....

-Mark

Apparently, this is known as Curry's Paradox.Well, there is a “Curry’s Paradox” indeed … but nothing to do with Paul Curry the magician.

See … http://plato.stanford.edu/entries/curry-paradox/

The geometrical dissection

trickattributed to Paul Curry is not aparadoxat all (though referred as such in many, many places … even the Stanford page above incorrectly calls it the “Geometrical Curry Paradox” … then again, they were just eliminating it as a possible source of confusion).Here, it is appropriately called for what it is … a dissection

fallacy:http://mathworld.wolfram.com/DissectionFallacy.html

gabe@XE> select atan(2/5), atan(3/8) from dual;

ATAN(2/5) ATAN(3/8)

---------- ----------

.380506377 .35877067

Deceptively close … but not quite.

Folks, you are missing two points here:

1- the dissection fallacy and other such examples apply to two different shapes arrived at with the same component shapes. Thats is not the case here: the two outside shapes are the same, apart from the missing square.

2- Like Tom says, there is no trickery: cut out the shapes and try it. There is no slight of hand or slight skewing or anything like that.

Just a consequence of using integer arithmetic to define areas that are not integer. A "rounding" error, if you prefer.

Noons,

Your point (1) is wrong - the two outside shapes are not the same, they only look the same because of a deliberately inaccurate presentation.

You point(2) is misleading - you don't need to cut them out, you only need to draw the figure accurately and you will see that the original is a lie.

Your last paragraph contradicts your point (2) - "using integer arithmetic" is exactly the "slight skewing" - one is the logical, the other the physical explanation of how the eye is being deceived.

This is, of course, a good analogy to the voodoo tuner who tells you that "I saw X, I did Y, everything went faster - trust me Y is a good thing". If X and Y are badly observed it's easy for the conclusion to be false.

Think about my demonstration of the /*+go_faster*/ hint - it's the same thing. I can SHOW you cases where putting this (pseudo-) hint into the SQL makes the query run faster - really - and if I don't describe accurately what I'm starting with you have to believe that the hint really works.

dunno, I just drawed the original in a piece of milimetric paper and it fit properly. Then the cutout from that missed the square when rearranged.

As for the voodoo tuning, I don't know: not familiar with the technique.

and now that I'm home, had dinner and coffee, I can perfectly see your point. Indeed, it's a deception.

They are likely taking advantage of the thickness of lines and the fact that eyesight fails to measure them properly when put next to each other.

The difference is very small, there is a hint of it in the hypotenuse of the large triangles and how it intersects the grid along its length. The result would be very close to the 5.2 and 7.8 you pointed out.

But I'm still trying to explain to myself how the yellow and green shapes can ever make a rectangle anywhere near 2x8. Which it most likely isn't anyway. By the time I've finished the Port I'm sure I'll have it figured out!

Thanks for pointing the fallacy out.

Suddenly this has started to appear all over the internet. I got atleast 4 email forwards ofcourse getting me stumped. Jonathan's explanation sets my confused mind to rest.

Okay, this is stupid. The triangles are not similar to each other. One is 2x5 and the other is 3x8. If you make the triangles congruent by changing 3x8 to 3x7.5, you have the same slope. Now, with this new adjustment, you have the second rectangular portion being 2x7.5 (15) which has the same area as 3x5 (15.. duh). So there you go. It's a stupid slope trick. Of course the stupid triangles are gonna be diferent if the slopes aren't the same. hmm.. that took like 5 minutes to figure out. What a stupid "paradox." It's just an eye trick.

As for the reason why the blank space is there 3x5 and 2x8 are 15 and 16. That space is supposed to be filled in. But they took it out for kicks so they could make a "paradox." Again.. STUPID STUPID STUPID.

What was stupid...

that you looked at it for five minutes or the "trick"? :) That space is not supposed to be "filled in", it doesn't work if it were.

Yes it is the slope.

Yes it is a trick of the eye.

Doesn't make it stupid - you cannot see the forest for the trees. It happens all of the time - we expected something and our eyes let us see what we expected.

what "slope" are you talking about?

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