Why does the triangle have extra uncovered area?

[RandallB]

Triangle with extra area?

Four puzzle pieces; two right triangles and two six sided polygons with all right angles.
The two six sided polygons can form a perfect 12 by 12 square as shown figure1.
They are put together in two different ways to create triangular shapes of exactly the same height and width (fig 1 & 2).
BUT, some extra uncovered area, a 4 by 4 square, shows up in fig 2!

*Where does this extra space come from?
*You can show you know the solution with one descriptive word.


Without a drawing function it's hard to show in 'text drawing' but with the dimensions given above and figures below it should be easy to make four paper cutouts to help solve if needed.

. . . . . FIGURE 1
0
000
000000
000000000
000000000000
0000000000000000
000000000000000000000
00000000000000000000000
222222222222222222222222II
222222222222222222222222III II I
222222222222222222222222II II II II I
222222222222222222222222III II II II II II
111111112222222222222222II II II II II II II III
111111112222222222222222III II II II II II II II II II
111111112222222222222222II II II II II II II II II II II II
111111112222222222222222III II II II II II II II II II II II II II
111111111111111111111111II II II II II II II II II II II II II II II II
111111111111111111111111III II II II II II II II II II II II II II II II III
111111111111111111111111II II II II II II II II II II II II II II II II II II III
111111111111111111111111III II II II II II II II II II II II II II II II II II II II I

. . . . . . .figure 2
II
III II I
II II II II I
III II II II II II
II II II II II II II III
III II II II II II II II II II
II II II II II II II II II II II II
III II II II II II II II II II II II II II
II II II II II II II II II II II II II II II II
III II II II II II II II II II II II II II II II III
II II II II II II II II II II II II II II II II II II III
III II II II II II II II II II II II II II II II II II II II I
11111111________2222222222232222222222220
11111111________222222222222222222222222000
11111111________222222222222222222222222000000
11111111________222222222222222222222222000000000
1111111111111111111111112222222222222222000000000000
11111111111111111111111122222222222222220000000000000000
1111111111111111111111112222222222222222000000000000000000000
111111111111111111111111222222222222222200000000000000000000000

 

[dextercioby]

SLOPE

THEY ARE NOT TRIANGLES

Daniel.

 

[RandallB]

SLOPE
Daniel.

WRONG
That does not describe the source the extra area.

 

[Pseudopod]

Sure it explains it, just hard to do it in one word. The hypotenuse of the red triangle must have a different slope than the hypotenuse of the black triangle. Specifically, the red triangle's hypotenuse must be a steeper slope than the black triangle's. This would make the first giant "triangle" (really a quadrilateral) have a concave hypotenuse and the lower "triangle" have a convex hypotenuse, accounting for the extra area.

 

[gerben]

here is a nice image:

 

[Gokul43201]

Not to be a pompous @$$ or anything, but since gerben's linked to that picture...
https://www.physicsforums.com/showpost.php?p=414521&postcount=20

 

[dextercioby]

Nice,Gokul,a page especially for u.Greg must have been in a very good mood that day... :tongue2:


Daniel.

 

[mattmns]

Off topic, but dextercioby you can press that number that is to the right of "View dextercioby's Warnings" for example #7 and it will go to a single post on one screen. https://www.physicsforums.com/showpost.php?p=444914&postcount=7

 

[RandallB]

Parallelogram

just hard to do it in one word. . . . quadrilateral)

Give credit to Pseudopod for coming close by mentioning the word quadrilateral.

The two large Triangular Shapes, easily mistaken for triangles, are of course quadrilaterals. Overlaying the small quadrilateral on top of the large quadrilateral will expose another quadrilateral shape as a visible portion from the larger .

Only this quadrilateral has two sides equal in length and both parallel to the hypotenuse of the small triangle and the other two are parallel with hypotenuse of the large triangle. Thus the best one word description of the source of the extra area:
PARALLELOGRAM

Which doesn’t directly give away the explanation,
but at least a less advanced poster was able to give the detailed description.

 

[Bartholomew]

I disagree. Someone who did not understand the problem would not be helped by the word "parallelogram"--paralellogram where? A better answer is "hypotenuse." That at least identifies which region.

 

[Gokul43201]

PARALLELOGRAM would be a poor choice of word. If you take any pair of congruent triangles (like what someone might interpret the two figue=res to be, if he hasn't cracked the problem), flip one triangle over, and stick it on the other, you get a parallelogram.

I prefer SLOPES (the plural being important), "HYPOTENUSE" (within quotes to indicate it is not really), QUADRILATERAL (this is straightforward and requires no tricks), or even SIMILARITY (the fact that the three triangles - the two real ones, and the bogus one - are not similar is a sufficient observation).

 

[RandallB]

SLOPES "HYPOTENUSE" QUADRILATERAL SIMILARITY

None of these directly describes “Where the extra space comes from” those are hints to or gives away the solution that a triangular shape is not necessarily a triangle. And all require more explanation, thus none was given as one word alone.

However to just “show you know the solution” implies doing so without giving away the solution to others.
Done to often by those that already know the answers and want to show off.

Yet once they do solve, they couldn't doubt someone clearly knew that had given the one word.
PARALLELOGRAM
Yet it does the least to give away the subtle misdirection of the puzzle.

PS: Gerben, thanks for the link

 

[K.J.Healey]

"Parallelogram" does not describe teh solution or problem at all... You DO know what a parallelogram IS right?

I think the best answer is "There is no extra area."

 

[Problem+Solve=Reason]

Ratio...

The hypotenuse length of the overall triangle (all shapes put together) is equal to the sum of the hypotenuse length on the II triangle and the base length on the 111 shape.

Because the 00 triangle is 4 spaces shorter in hypotenuse length and height (as seen) than the II triangle, the 4x4 block must be inserted for the pieces to fit correctly, yielding the same measurements as the first whole triangle.

____________________________________________
In seeking wisdom thou art wise; in imagining that thou hast attained it - thou art a fool.
Lord Chesterfield

 

[RandallB]

"Parallelogram" does not describe teh solution

Never said it described the solution! It describes how figure 2 is larger than figure 1.
Figure 2 has to outline a larger area than figure 1 to allow for the 4x4 hole.
Did you and Lord Chesterfield try cutting out the four shapes?
Your both wrong, just trace them on paper if you have trouble seeing it.

 

[K.J.Healey]

Never said it described the solution! It describes how figure 2 is larger than figure 1.
Figure 2 has to outline a larger area than figure 1 to allow for the 4x4 hole.
Did you and Lord Chesterfield try cutting out the four shapes?
Your both wrong, just trace them on paper if you have trouble seeing it.

What do you mean I am wrong? I understand completely where the hole comes from. Neither of these two shapes are triangles, I don't see why people would have any trouble seeing this at all, unless the image that shows it is distorted or misleading.

 

[RandallB]

"There is no extra area."

So your saying you finally did find the Parallelogram?
You DO know what a parallelogram IS right?

Follow the link for a better photo, but if you need to SEE the Parallelogram
cut out the four shapes.

 

[Integral]

A parallelogram, by definition has parallel sides that shape is NOT a parallelogram. Nor is it a Triangle. It is a Quadrilateral, It has four sides none of which are parallel to any other.

 

[Gokul43201]

A parallelogram, by definition has parallel sides that shape is NOT a parallelogram. Nor is it a Triangle. It is a Quadrilateral, It has four sides none of which are parallel to any other.

RandallB never claimed that the provided shape wa a parallelogram. I think what he said was that if the two shapes are superimposed, the bigger one will have an excess area that is the shape of a parallelogram. This excess parallelogram has the same area as the removed square.

 

[K.J.Healey]

RandallB never claimed that the provided shape wa a parallelogram. I think what he said was that if the two shapes are superimposed, the bigger one will have an excess area that is the shape of a parallelogram. This excess parallelogram has the same area as the removed square.

Ahh, I didn't follow what he was trying to say. I thought he meant somehow it all had to do with laws of parallelograms, not that it would generate that shape... I see now.

 

[Ubern0va]

If you want to physically confirm that some outside block has been pulled into the triangle by differences in slope, complete the rectangles using the empty space that corresponds to the two smaller triangles. Then complete the rectangle on the larger triangle and you will see that there are 16 blocks left outside the upper resulting shape and only 15 for the lower one (the 16th required block is inside the triangle).

 

[RandallB]

Wow Ubern0va you've dug an oldie. but one of my favorites:
Lots of folks answered this one to quick.

The point of the OP in this one is to show you understand the problem of how two “triangular shapes” of the same size (height & width) can have different areas.
If they really are triangles then they should have the same area

To show you understand you only need demonstrate that you know:
“ *Where does this extra space come from?”
And you can prove that know in one word because:
“ *You can show you know the solution with one descriptive word.”

So if your trying to give a solution, (I have no idea what your saying in those several lines) you only need give one and only one word to describe the shape of the source of that extra area.

If you want to keep trying without looking at the other posts. I’ll let you know when you get it right.

BUT here’s a tip, if you just ‘think’ you have the word; you likely do not.
Because when you do get the word you will have no doubt.
When you have no doubt, just browse though the rest of the thread.

 

[Ubern0va]

Actually I've already come to the realizations that are listed in this thread; I was just adding a simple way to confirm them, a way to "see" the extra area, for those who don't have the ability to approach this mathematically, or still aren't convinced for some reason.

 

[RandallB]

Actually I've already come to the realizations that are listed in this thread; I was just adding a simple way to confirm them, a way to "see" the extra area, for those who don't have the ability to approach this mathematically, or still aren't convinced for some reason.

Then you know discribing something as a "block" in "(the 16th required block is inside the triangle)" dosn't go to the solution right.

 

[Mk]

I always thought Gokul's post settled it, but hey I guess not.

 

[Hurkyl]

Well, the problem is that what's "obvious" often ain't. RandallB has fixed in his mind what he thinks to be the crystal clear unique answer to the problem he posed, and he wants to insist to everyone else how crystal clear it is. :grumpy:

But I agree with Healey's reaction -- it's such an obscure way to answer the problem that I too would be more likely to think someone who used RandallB's word is totally clueless, rather than thinking they had the solution.

 

[RandallB]

Well, the problem is that what's "obvious" often ain't. RandallB has fixed in his mind what he thinks to be the crystal clear unique answer to the problem he posed, and he wants to insist to everyone else how crystal clear it is. :grumpy:

But I agree with Healey's reaction -- it's such an obscure way to answer the problem that I too would be more likely to think someone who used RandallB's word is totally clueless, rather than thinking they had the solution.

Balderdash how could anyone accurately come up with the correct answer, parallelogram, by accident and be clueless?
Believing the best answer is "There is no extra area" would be clueless.

No without having to hear any explanation at all, I know anyone that accurately said parallelogram understands the problem completely.

Complaints to the contrary are just sour grapes over not figuring out the whole solution on their own.

 

[theonerester]

im still stumped because the slope doesn't seem to be that significant from the original image to make way for enough space to fill up one square

 

[RandallB]

im still stumped because the slope doesn't seem to be that significant from the original image to make way for enough space to fill up one square

Trust me “theonerester” understanding this puzzle COMPLETELY is not as easy or crystal clear as some may claim.
Your understanding it is simply a matter of completely visualizing it and just taking some time with it. Most modern scientists seem to me to get stuck in “Mathematics” mode and quite looking at a problem, once they come up with any mathematical way of describing something.
But, an equally important part of science is “Modeling” – it was after all the key to understanding “DNA”.

So be a scientist willing to get out of the ivory tower and get your hands just a little dirty. Actually, use paper and scissors, even if you think it seems like grade school, it’s called Modeling!
Really, draw and cut out the four figures.
Use them to build and trace the two shapes in question.

Since one shape covers a larger area by virtue of having a hole in it, the other shape should fit inside of it – so trace it within the larger tracing.

You should see three quadrilaterals, one being the solution you’re looking for.

Once you see it --- geometry will see you the rest of the way to put a name to it and define its area.

 

[davee123]

Complaints to the contrary are just sour grapes over not figuring out the whole solution on their own.

Wow. That's pretty insulting.

I think the problem is that you're so very sure that that's the one and only answer, and seem unwilling to accept anything else as a possibility. I would argue that stating "quadrilateral" is equally accurate of a response, just slightly less specific.

Some answers, such as "slope", "bend", "angle", "convex", "concave", "dissimilar", "ratio", etc., show that someone understands the implied paradox, but I agree, doesn't necessarily answer the "where" that you asked.

But really, "paralellogram" isn't the answer to the question you asked, either. You asked "where", and the answer you wanted was "what shape?". Assuming the 90 degree angle of the large "triangle" is placed on the X,Y plane, with the shorter side along the Y axis, and the longer side along the X axis, the answer is actually more like:

"The 'extra' area in the second diagram described by the polygon formed by (5,0), (5,1), (6,1), (6,0) is redistributed from a polygon in the first diagram described by (0,5), (8,2), (13,0), (5,3)".

That actually describes "where" very exactly, but I'll be damned if I can sum that up in one word, especially since it requires the definition of the placement of the two diagrams on the X,Y plane.

Some other one-word answers that I think might be acceptable would be:
- Hypotenuse
- Diagonal
- Northeast (if as diagramed above on standard X,Y plane)

And stating "quadrilateral" is probably just about the same as saying "parallelogram", excepting the fact that they might be attempting to describe the flaw in the "triangle" rather than describing the area that the "extra" space came from, so it's difficult to tell if they were giving the right answer or not.

So, I probably would've accepted those three answers, plus "paralellogram" and "quadrilateral", and maybe others.

But regardless, I wouldn't be so quick to tout a specific, correct answer in questions like this. Anything that's this open-ended is likely to have a wide variety of answers which are arguably correct, since an actual answer isn't possible within a single word.

DaveE

 

[RandallB]

But really, "paralellogram" isn't the answer to the question you asked, either.

Get a life for crying out loud it is a brainteaser (an old one at that with a fresh twist) not religion.

Are you saying you already understood it so well you knew that the small shape could increase in size by pushing the false concave ‘hypotenuse’ out to a convex one, with area bounded by those two false ‘hypotenuses’ exactly matching the empty square that appears inside the larger shape. And you already knew and understood that space well enough to describe it and you’d pick something other than "parallelogram" to name it.

I don’t think so. I’m sure not convinced.
And no, I don’t mind calling Sour Grapes for what they are.

What’s with you and others -- What are you going to do if someone comes up with something important like a truly new theory that legitimately displaces your favorite theory? Say thanks and congratulations for finding it, or complain that you didn’t come up with it first?

 

[davee123]

Get a life for crying out loud it is a brainteaser (an old one at that with a fresh twist) not religion.

Right back atcha :)

Are you saying you already understood it so well you knew that the small shape could increase in size by pushing the false concave ‘hypotenuse’ out to a convex one, with area bounded by those two false ‘hypotenuses’ exactly matching the empty square that appears inside the larger shape. And you already knew and understood that space well enough to describe it and you’d pick something other than "parallelogram" to name it.

I don’t think so. I’m sure not convinced.

I'm not sure I understand-- you're still maintaining that the necessarily unique solution in one word is "parallelogram"? I agree that it's what I would consider to be an acceptable answer, but I would also accept others. Perhaps you could elaborate on why "hypotenuse" is necessarily incorrect? Or perhaps "hypotenuses", which together would describe the parallelogram you were looking for?

As I elaborated in my previous post, "parallelogram" is an incomplete explanation, and I would argue that it's likely that ANY one word solution is similarly incomplete. Hence, I would maintain that in the absence of a complete solution, a either a partial solution must be acceptable, or none must be.

However, I think you're approaching the problem strangely emotionally, which is why I commented on the fact. You appear to be overly proud of your particular solution, and are a bit too eager to discount other people's interpretations and claim they lack understanding of the problem, when in fact, their interpretation may just be different than your own. And I think that was pretty insulting of you. Long story short, I would advise that if you're going to come up with a brain teaser, make sure that either:

1) You have a well-defined solution, allowing you to disprove incorrect answers.
2) You're open to other possible interpretations apart from your own.

DaveE

 

[RandallB]

Perhaps you could elaborate on why "hypotenuse" is necessarily incorrect?

In no way can the word discribe something that has an area.[/QUOTE]As I elaborated in my previous post, "parallelogram" is an incomplete explanation, [/QUOTE]You mean where you detailed the location of a polgon that can only be completely discribed as a parallelogram (defining an area or "space")?

All I was doing was correcting incorrect interpretations based on the brainteaser given. Emotional reactions like yours seem way to unproductive to waste time on – maybe I should just ignore them even if it does leave incorrect assumptions uncorrected and trust the more attentive to get the point on their own.

I’ll unsubscribe from my thread, do as you please.

 

[davee123]

In no way can the word discribe something that has an area.

Think of it as though it were an animation. Going from diagram 1 to diagram 2, the "hypotenuse" of the large triangle goes from being concave to convex. The area that is contained within the change of the hypotenuse is the same area that's described by the parallelogram. Hence, by viewing the hypotenuse as a discrete, flexible element that changes with 'time' over an area, "hypotenuse" describes the location and the area just as well as "parallelogram" does.

In fact, I might argue that human-understainding-wise, "hypotenuse" might be a better answer, because the hypotenuse of the large triangle is already clearly drawn on each diagram, hence it's already been established and can be examined as a potential answer. The "parallelogram" you reference, however, isn't clearly drawn on the diagram, and as such isn't as immediately clear of a concept.

Emotional reactions like yours seem way to unproductive to waste time on – maybe I should just ignore them

Why wouldn't you? For that matter, why did you bother making the assesment "Complaints to the contrary are just sour grapes over not figuring out the whole solution on their own"? You're quite intentionally pointing out other people's emotional reactions and branding them as negative, as well as implying their stupidity, more directly by declaring that they lack understanding of the problem. I think it was a waste of your time to point that out, and only served to hurt other people's feelings (probably making them feel defensive), and (as I'm trying to point out) could be entirely inaccurate.

The only reason I'm pointing it out here is to try and get you to realize that the comment (the attitude really) was unnecessary and divisive. I'm making the point in the hopes that you won't do such things in the future. It doesn't help to the understanding of the brain teaser in question, and only serves to make a more hostile environment in the forums.

If you think someone has an incorrect solution, try and either explain why it isn't correct or have them try to explain why it *is* correct. Please don't just assume that they're clueless and treat them as such.

DaveE

 

[NateTG]

I think that a better explanation would be:

Neither of the shapes are actually triangles, even if we fill in the 'missing square'. This puzzle is based on an opticall illusion that makes you assume that the 'hypotenuse' is a straight line when, in fact, the 'hypotenuse' is bowed in in one geometry, and bowed outward in the other, and, moreover, the difference in areas between that covered by the bowed and unbowed hypotenuse is exactly equal to the 'missing square'.

(As soon as people see that the shape/hull in question is not a triangle, but a quadrilateral, the rest is very easy to understand. Notably, this optical illusion is especially insidious because we're trained to use sloppy sketches as triangles in math.)