How can we explain the triangle paradox on this riddle website?

[Edgardo]

Check out this riddle:
http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/jigsaw-paradox.html

Where does the hole in the lower figure come from?

 

[lawtonfogle]

easy

when rearanged the triangle area, counting the square, is bigger. I'll try to explane this by a step by step post. The bottom left of the graph be the origin.



notice all we are adding is just 1 square unit. 1 is not that big and when split and added to 25 other units, it can be hard to notice the change, expesily if grouped together.

( u might be like this guy right now so let me cleat it up with an example)

an example is,

a 10 by 10 square on the graph is increased in size so that instead of 100 area it has 101 area, the change can be hard to notice, very hard if you don't know what to look for.

( i hope your like this guy who understands it now)

 

[lawtonfogle]

(pull up the picture and make it small, it will be helpful)

look at (6,11), there is a little space to the SE of it

now look at (6,3), that space is not there . actualy, there is some space up to the NW. one might say the figures are not drawn to scale, yet if they were it would be the same problem. This is the 1 area that is split and added to all the edges. The area, if measured from a triangle drawn to scale, would be 1 bigger for the lower triangle than the upper one, if you measured 'EXACTLY'.

 

[RandallB]

( i hope you understands it now)

Not from that "Easy" Explaination? Can't be true because as the prolem states all four parts are the same!

To show Edgardo you "get it" tell him the NAME of the source for the "extra" space in one word - just one.

Hint in white:
The word is a spiecal case for the correct name of the two "Large triangles"

 

[Telos]

I've never heard of "Fibonacci Bamboozlement," but that first "triangle" is actually a quadrilateral. There's a barely perceptible "dent" in the first figure and a "bump" in the second.

You can tell because the slopes of the two smaller triangles are different (2/5 green, 3/8 red).

 

[Doctordick]

I've never heard of "Fibonacci Bamboozlement," but that first "triangle" is actually a quadrilateral. There's a barely perceptible "dent" in the first figure and a "bump" in the second.

You can tell because the slopes of the two smaller triangles are different (2/5 green, 3/8 red).

You're right on the money. You show strong signs of being able to think!

Have fun -- Dick

 

[RandallB]

that first "triangle" is actually a quadrilateral.

So what do you call the 2nd "triangle"??
And back to the original question - where did the "extra" space come from??
You should be able to define it in just one word.

Keep thinking you'll get it.

 

[Edgardo]

I would day that Telos gave a good answer (I had the same thought).

 

[RandallB]

I would day that Telos gave a good answer (I had the same thought).

I assume a good answer is good enough for you or you haven't figured it out yet. Another hint - cover the "Large triangle" with the "small triangle" and trim off any exposed part of the large.
Now discribe the triming.
And no " quadrilateral " will not do.

 

[Telos]

Randall, the "hole" comes from a rearranging of the shapes.

There is no lost area, so there is no hole.

 

[RandallB]

Randall, the "hole" comes from a rearranging of the shapes.
There is no lost area, so there is no hole.

What you think the triangles are the same size?
Obviously you didn’t do what I’d suggested - so why are you commenting? Use a real pair of scissors cut out four real “shapes” and trace out two “triangles”. Then really trim out the “extra” space on the larger using real scissors and you can actually have it in your hand. Then get back with us and tell us what it is.

 

[Telos]

Randall, when you said, "large triangle" and "small triangle" I thought you actually meant the real green and red triangles - because, neither of the two larger figures are triangles. So your statement was very confusing. I meant no offense by not responding to it. Moreover, this is a math puzzle and this is a math forum. And we're not kindergarteners. We should be able to figure this out without resolving to construction paper.

green triangle area => (1/2)2*5 = 5 units
red triangle area => (1/2)3*8 = 12 units
yellow polygon area => 7 units
light green polygon area => 8 units

total area for both figures => 5+12+7+8 = 32 units

The area is the same for both figures. There is no lost area. There is no missing section.

Now, if I'm making a mistake, please tell me. And stop keeping us in suspense, let us know the word you're thinking of. Is it "tangram?"

 

[RandallB]

Randall, when you said, "large triangle" and "small triangle"

There in (" ") because as you know there are NOT triangles!

We should be able to figure this out without resolving to construction paper.
total area for both figures => 5+12+7+8 = 32 units
The area is the same for both figures. There is no lost area.

Of cource you shouldn't but sometimes using somethink real helps get your abstract mind to work better!
NO they are NOT the same -- If needed turn the rectangle over created by the Polygon’s so the space is in the middle of the shape! The larger one is larger! Why? one word will do!

stop keeping us in suspense, let us know the word

First get you math right, if you say you cut it out and still don't "get it" then sure I'll tell - but when you do get it you'll look back on this thread to find a massive number of hints. Working brain teasers is about learning from the doing not having answers given to you!

 

[Telos]

The larger one is larger! Why? one word will do!

"Tautology?"

You want me to cut out triangles and squares to help my abstract mind work better? These are triangles and squares. You're insulting me and wasting my time.

You've just been added to my ignore list.

 

[Uno Lee]

It took me a while but here is what I got Large triangle slope =3:8
Small triangle slope=2:5
lowest common denominator = 40 Large triangle slope 15:40
Small triangle slope 16:40
the slopes are different: we assume the complete structure is a triangle but it isn't. It is actually a four sided object, including the "missing square." Another way to put it is the space taken up block in the basement is made up for by flipping the roof so the water doesn't pool. You got to look at this whole thing on a bit of an angle.

 

[RandallB]

It is actually a four sided object, including the "missing square."

Pretty good Uno Lee at least your thinking, as you’ll note earlier in the thread the word you’re looking for to describe the “triangle” is quadrilateral. And the extra area inside the larger is also a quadrilateral! A special case quadrilateral called a square! So the question remains where did the area displaced by the square go to in the larger quadrilateral? (trust me this is simple stuff) As also hinted at earlier it shouldn’t be a surprise that the answer is a quadrilateral, but that would the best description of it. (Just like no one calls a square a quadrilateral even though it is.)

The thing I like about this puzzle at it so many people that “know so much” quickly get this and move on from it without completely “getting it”. When they do they would immediately know this word and why it’s important, and how it fits so well with what you already know. Then you’ll know you “completely get it”.

This process of learning how you yourself learn and discover, within yourself is a valuable thing and I won’t rob you of it by just telling you the answer. Trust me developing that kind of personal skill will help you in the future with ideas like Relativity, Quantum…, or whatever without having to depend on others to think for you.

But, if you really want to give up the easy way to find the answer since this Forum is full of old stuff like this! Just “Search This Forum” under brain teasers for triangle.
It's pretty simple and since I know the answer I’ll let you post it in this thread. Just let us know if you looked it up or figured it out.

 

[AC]

similar triangles
the 2 triangles are supposed to be similar.
but 2,5 and 3,7 small squares are not valid

 

[Moo Of Doom]

Certainly the word RandallB is thinking of is parallelogram?

 

[T@P]

this is probably the most "popular" puzzle I've seen on these forums... really its kind of ridiculous now that I am seeing it for the 87th time. :zzz:

 

[RandallB]

Certainly the word RandallB is thinking of is ...?

Your correct - You'd be surprized how many smart ones "Know" this Puzzle - so they snooze through it without completely recognizing this source area for the more easily seen square area. Yet even the most novice can find it with a bit of diligence.

Thanks for keeping it in white for those new to the forum wanting to work it on their own.

 

[pack_rat2]

The smaller triangles aren't similar, but they're close enough so that you think they are, and you think they're similar to the big triangles. If you place the edge of a piece of paper along the hypotenuses of the the large triangles, you notice the upper one sags, and the lower one bulges.

 

[pack_rat2]

"Below the four parts are moved around. The partitions are exactly the same as those used above."

Indeed they are...no fibbing here. We tend to assume the triangles are similar because on first glance, they certainly seem to be.

 

[niranarch]

There is no optical illusion. Its just a careful manipulation with line thickness checkout this drawing i made in CAD ( you can try for yourself)

http://sphotos.ak.fbcdn.net/hphotos-ak-snc3/hs164.snc3/19161_340546608708_687698708_5005153_549897_n.jpg

 

[davee123]

There is no optical illusion. Its just a careful manipulation with line thickness checkout this drawing i made in CAD ( you can try for yourself)

Although yes, you could do that, it's not what's going on in this problem. In the problem, there are 4 shapes:

A) Teal right triangle measuring 2 units by 5 units
B) Red right triangle measuring 3 units by 8 units
C) Orange irregular hexagon with area of 7 square units
D) Green irregular hexagon with area of 8 square units

These shapes don't change, and neither do they overlap. In your example, you'll notice that you've drawn shape (A) with a height slightly less than 2, and shape (B) with a height of slightly more than 3, so the shapes are misrepresented in your version. The OP has nothing to do with line thickness whatsoever-- you can see this for yourself if you'd like to make a similar CAD drawing with the shapes as defined above, with a miniscule line thickness.

DaveE

 

[Nick89]

Haven't read the whole thread but the answer is really simple:

Spoiler

The red and dark green triangles don't have the same slope.
So the total shape is not a triangle at all, it merely looks like one to the naked eye.
If you draw a line along the slope of either triangle, you can clearly see it, as in the picture below:
Proof!

 

[niranarch]

@dave123- have you tried drawing it? its impossible to draw as you see in the original one.
If you are trying do try it in a CAD software so that thickens of lines don't effect you. In a small drawing line thicknesses seem to be irrelevant but can easily hide such differences

 

[davee123]

@dave123- have you tried drawing it? its impossible to draw as you see in the original one.

No, it's not impossible. Heck, it's right there in the very first image. I think the problem you may be having is that you're attempting to draw it by STARTING with a triangle, and then subdividing it into 4 sub-parts, which is correctly impossible. Don't do that. Start with the 4 sub-parts, and arrange them as shown. Arrangement #1 doesn't have a square-shaped gap, and Arrangement #2 DOES have a square-shaped gap.

Spoiler

The trick is that it's NOT a triangle. So if you start by drawing a triangle, then try to draw the relevant sub-shapes, you're doing a different problem, and you've been fooled! Instead, notice that the slopes of the two small triangles are NOT the same-- so the resultant shape isn't a triangle at all! The reality is that the square gap on the bottom is replaced by a very narrow, thin gap all along the "hypotenuse" of the large triangle.

DaveE

 

[aidanapc1359]

OK Guys,

First of all the problem states that the partitions in both large triangles are all equal.
So please just accept this. No measurements are necesary or relevant.
Also the the two large triangles are equal.
The problem arises due to topology.
or simply "there is no conservation of area or volume".
With a 2D shape you can maintain area while changing perimeter.
For example, two rectangles a 1m x 4m, area = 4 m2,
For a 2m X 2m rectangle, area also equals 4 m2
However the two rectangles have different perimeters!, The first = 10m, the second = 8m
So by changing shape you can maintain area and change perimeter.
This is similar to the 3D version of volume and surface area.
The reason why your small intestine has so many villi and microvilli in it, is because it greatly increass the surface area for absorption.
Also why a piece of sodium reacts much slower than powded sodium (crushing into a powder greatly increases surface area without changing volume.
Getting to the point:
Basically because the partitions no longer slot together in the bottom diagram, this has changed the shape but maintained the area (however the effective area for covering has decreased and created a hole)
Thus the perimeters of the two figures are different!
Another way of answering this is because the overall shape is now irregular it can no longer cover all of the area of the bottom diagram, even though there has been no loss of area in the partitions!
Forget about measuring squares, this problem is not about deception but mathematics.
Hope this makes sense.
By the way, this is my first posting, very pleased to meet you all

Aidan.

 

[nashsur]

I like to keep things simple and maybe that is my mistake. To me it is just the difference in length of the orange and green thinner sections.
Please note the overall length of the red triangle is 8 blocks. There is only one way to combine the length of the orange and green blocks to make 8. However, the thinner sections of the two blocks are of different lengths. One is 3 and one is two. No matter what you do to combine them to an overall length of 8, you will have the resulting hole. That is the long and the short of it. It seems to me that if you are trying to create a base for the larger red triangle it would be to your advantage to create a base that is the same length as the red triangle. To that end you would combine the orange and green blocks in a configuration that would total an equal length of the red triangle. I am sure there is a mathematical expression to describe this but it is not needed here. Other than to say 4 + 4 = 8 and 5 + 2 = 7. That leaves 1 not accounted for. Why add all these other complications to it?

To wrap up; overall length is red triangle base plus green triangle base. Green triangle, orange block and green block are of equal base length. Red triangle base length = 8 . All other shapes base length = 4. Red triangle is 3 high. All other units are 2 high on one end. The two block shapes have one end that is 1 high. The 1 high end on the orange block is 3 longer than the 2 high end and the 1 high end on the green block is 2 longer than the 2 high end. The only way to combine the two blocks to equal the correct length of the red triangle and the height of the green triangle is to put the two 1 high ends of the orange and green blocks on top of each other so as to cause the overall height to be 2 and the overall length of the combined orange and green blocks to equal 8. This will automatically leave a space between the orange 2 high level and the green 1 high level because 4 + 2 does not = 8.

Again, if I miss the point of the exercise please let me know.