By Vurdlak on March 19, 2006, with 207 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.
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its the triangles if you cut the red triangle right in the middle to match the green one you’ll see that the green one is bigger. By switching them around you can create a steeper slope of the line or a flatter slope. This is how the extra block is made. Since the green triangle was on top of the red it caused it to take more space because of its steepness and caused no extra space. And when you switch them around there is extra space.
conjunction
I don’t know if it matters, but the slopes on the red and green triangles are slightly different. The green one is 2/5 and the red one is 3/8…
The hole come from the rearrangement between partion green and partion orange.they both have 5 cubes,but partion green has 3 cubes and partion orange has only 2 so when rearranged it creates a hole.
this is too easy to solve
first picture look at coordinate 6,10.9
second picture look at coordinate 6,3 – notice the small tip difference
the difference in that point makes the first picture coordinate = 6,10.9, instead of 6,11
for the first picture, when the y point is 11, you’ve to move x to 6.5 > so that extra 0.5 extends the length of red triangle
The colored shapes are the same in the two cases. What is not obvious, and this is explained by MuntainDW, is the shape that the parts make. The two shapes are not triangles, OK? It’s an illusion, that’s all. The two shapes are not right-angled triangles because the hypotenuse is not a straight line. To be a straight line, the red and green triangles have to have the same left side (sharp) angle. For that, the tangent (or slope) can be verified:
- RED triangle tan=3/8=0.375
- GREEN triangle tan=2/5=0.4
So they determine a concave (inward) “hypotenuse” in the first case and a convex (outward) “hypotenuse” in the second case, the area being slightly larger than the first.
Both of them are not triangle. they are polygon.
This case proves the Gestalt Theory.
People try to get things in a whole by form
If QI is right, then the edge of the green and red triangle is actually sloping. A French magician invented it – can’t remember which one, though.
The curves of the triangles are not exactly the same. They slope inwards on the top, but outwards on the bottom. You can tell at the red tip on the bottom picture.
lol, my girlfriend ask me today about it, and i make my own proven on corelDraw.. and the answer is, the hypotenuse is not the straight.. here my proven
sorry here’s the image:
I think slope is not a factor here…..the upper green triangle is not actually a triangle its a quadrilateral….here…
http://imageshack.us/photo/my-images/21/e001f.jpg/
WTF I AM GETTING THIS WEIRD GLITCH. EVERY TIME I CLICK RANDOM ILLUSION IT COMES UP WITH THIS ILLUSION! WEIRD!
The key to the puzzle is the fact that neither of the 13×5 “triangles” is truly a triangle, because what would be the hypotenuse is bent. In other words, the hypotenuse does not maintain a consistent slope, even though it may appear that way to the human eye. A true 13 × 5 triangle cannot be created from the given component parts.
The four figures (the yellow, red, blue and green shapes) total 32 units of area, but the triangles are 13 wide and 5 tall, so it seems, that the area should be S=\frac{13\cdot5}{2}=32.5 units. But the blue triangle has a ratio of 5:2 (=2.5:1), while the red triangle has the ratio 8:3 (≈2.667:1), and these are not the same ratio. So the apparent combined hypotenuse in each figure is actually bent.
The amount of bending is around 1/28th of a unit (1.245364267°), which is difficult to see on the diagram of this puzzle. Note the grid point where the red and blue hypotenuses meet, and compare it to the same point on the other figure; the edge is slightly over or under the mark. Overlaying the hypotenuses from both figures results in a very thin parallelogram with the area of exactly one grid square, the same area “missing” from the second figure.
According to Martin Gardner,[1]the puzzle was invented by a New York City amateur magician Paul Curry in 1953. The principle of a dissection paradox has however been known since the 1860s.
The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers. Many other geometric dissection puzzles are based on a few simple properties of the famous Fibonacci sequence.[2]
Here’s a solution to this conundrum:
Great job Mysteryman25!
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase. In simple terms, the variation in space of any quantity can be represented (e.g. graphically) by a slope. The gradient represents the steepness and direction of that slope.
The green triangle has dimensions 2 x 5 and gradient 2 / 5 = 0.4
The red triangle has dimensions 3 x 8 and gradient 3 / 8 = 0.375
Hence the gradient of the green triangle is greater than that of the red triangle.
An exaggerated outline might look like….
In summary the missing square in the bottom triangle is made up for by the fact that it’s hypotenuse bends out where as for the top triangle it bends in.
thanks to Mysteryman25!!! now i got it
Thank u Mysteryman25! Wish u were my math teacher lolz!
Thank you Mysteryman and all the others who gave this one away to me. I just COULDN’T figure it out, and it was driving me nuts, causing me to question my sanity and the basic rules of geometry. Now I can sleep peacefully again.
wow, mysterman, you are mathematician, now i got it
tan (2/5) tan (3/8), so the hole triangle is not realy a triangle
this is the correct drawing:
the hole really exist but triangle right triangle 13 x 5 doesn’t…
Same shape different area placed
Iget it now from that moving example. thank you mysteryman25. its not so confusing now!!
MYSTERYMAN25 — thank you!
It took about 8-12 aec’s for me to see it. My mind works in pictures and math. Still fun! Thank you!
Glenn
I gave up on this one a long time ago. Thanks to Grant and Mysteryman25, now it makes perfect sense!
Wait, what about Mr. Illusionman.He can be your math teacher. He get the solutiion before MysterMan25. Not fair :(
Please, somebody. Support illusionman. He’s smart. He can be your math teacher. If you support mysteryman you are math cheater.
Illusionman is indeed smart, too smart I guess for ordinary people like the majority including me. What most people need to learn from a Math teacher, and from any other subject for that matter, is to make the solution easy to comprehend, and in this case, visualize.
Hrm.. interesting ideas, but, i just did this with graph paper, and it works. there is a new square out of no where.
I think most of your explanations are wrong
as i painted them on a paper and they are identical .
There is no missing parts or gimmicks :o
this is easy….the stupid git has drawn it rong…thiko
The original text lied, mysteryman is right they are not the same size. I reopened the image in photoshop and using the pen tool drew straight lines along the hypotenuse. It is slightly different in each drawing, bowed out slightly and bowed in slightly, giving the extra space for the ‘hole’ I see it, understand the animation, but it still doesn’t make sense somehow.
The acute angles in the red and green triangles are marked as equal when they are — in fact — not. If the large triangle actually existed, the two angles would be the same, but they’re not and it isn’t … It’s an optical illusion. A good one, but still an illusion.
Bill (the author of a math book)
It says dont give the answer away. All of your explanations are just confusing me further.
Then again, math was never my strong point.
Mystery Man 25 – very elegant answer – how did you do the animation? Great job!