The picture is puzzling. Found on the web.

<a href=http://www.scubateamuden.net/forum/viewtopic.php?t=166>LINK</a>


I wanted to put the picture here but upload didn't work properly..

Sorry.


(Solution kan be obtained from the Site, but is not published (yet)).

Originally posted by ProCop
The picture is puzzling. Found on the web. Have not the sollution.

very puzzling

Thats better theres a link now!

what is puzzling?
the triangles are not the same.

In fact, you aren't technically dealing with one triangle in either picture.

The two smaller triangels each have slightly different angels.

Edit: Hell, look at the two big triang;es even. They aren't the same. They have different areas, that is why the squares don't all fit.

If you count the squares forming the triangle(s) they are exactly the same (on both pictures) - please check. (smalle triangle 2/5; bigger 3/8) I am puzzled...

that is the problem, you are counting squares. look at where the hypotenous crosses the grid on both of them. they aren't teh same tringles, or those points would be the same too.

OK they differ (a very little). Still, it seems suprizing that you can get such combination, I wouldn't judge, at first, that it would account for a whole square...

PS. the area of the big triangles is in square-sides the same:13/5

the deal breaks down to this: the small triangle isn't similar (in the geometric sense) to the large one. They have different angels. That is why the points don't match up. It they where, you would probably have to chop up some of those other squares to fill in the shaded area when you reversed them.

What you have is four peices that, when put in one order form one 'triangle' (and i use that loosely because the angles of the two smaller ones are different). When put in a sllightly different order, the area under the larger hypotenous is larger.

the area would be the same if the angles where the same. they aren't..

I cut the picture in PAINT and compared the small squares. They seem the same. But I read your argument. It is in the small nuances.

Are all the squares the same size? Maybe the lower triangle is made up of slightly smaller squares or slightly less tall squares or something.

It seems to me that there cannot be a difference a difference in area if the sides of a right angle triangle are the same. As such, if the lower triangle has more squares the average square size must be smaller.

Does anyone speak Finnish or whatever? :)

Cheers!

editted for clarity

It's a Dutch site. I do not really understand the explanation. (Not born Dutch speaker)

Edit: but I will translate it Sunday and post it...(they have a mathematical explanation but it must be asked for by email)

if i compare the ratio of sides of the triangles

smallest = in squares 2/5 = 0.4
middle = in squares 3/8 = 0.375
big = in squares 5/13 =0.3846....

that's I think what makes the differce.

Due to the different angles of the two smaller triangles, the sloping edge of the large triangle is bent in the middle. In the first picture the bend is concave (inwards) and in the second convex (outwards). This means its area changes.

In the two pictures the area of the big triangle is:
First picture: 5*2/2 + 3*8/2 + 3*5 = 32 squares
Second picture: 5*2/2 + 3*8/2 + 2*8 = 33 squares

I beieve Mephura got it right. Observe where the each line crosses through in the outer edges of the two triangles. You'll notice the top one is smaller but if seen quickly (as it was meant to, hence the puzzling part) it doesn't really show through. The viewer if left thinking that the two triangles are same when they differ.

More succinctly, the triangles appear, at first, to be identical 5 x 13 right-angled triangles, with areas of 32.5 square units each.

The first "triangle" is actually only 32 square units, half a square unit less than a true 5 x 13 right-angled triangle.

The second triangle is 33 square units, half a square unit more than a 5 x 13 right-angled triangle.

Hence the difference of one unit between the two triangles.

damn i is good..
actually, you guys are the ones that figured out the hard stuff...
I just noticed they weren't the same is all.

The first "triangle" is actually only 32 square units, half a square unit less than a true 5 x 13 right-angled triangle.

The second triangle is 33 square units, half a square unit more than a 5 x 13 right-angled triangle.




The small and middle triangle are <b> absolutely</b> equal, therefore the difference is not in triangles (or half a squares) but in the <b>whole</b> squares.

What?

What?

The diffrence is not in the fact that some squares are only partialy filled. The small triangles are identical ---> the partialy filled squares are identical....it is <b> not </b> the partially filled squares where the diffrence remains: please re-check and re-consider.

Surely, what I said was accurate? You're not making yourself very clear.

Well if you recognise (and meant) that the only difference which is to be found in the two (original big squares) is <b>on the level of squares only </b> and not on the level of parts of squares, then it's OK and I misunderstood you - my apology for that.

Well, to avoid confusion, I'll write a more lengthy explanation of what I said earlier:

The area of a right-angled triangle is &frac12;<i>ab</i>, where <i>a</i> and <i>b</i> are the two sides opposite the hypotenuse. The two triangles appear to be identical right-angled triangles of area

&frac12;(5*13) = 32.5 square units.

But they're not. If you sum the areas of the shaded regions, you find the top triangle is only 32 square units in area. This is due to the aforementioned concavity of the hypotenuse of the "triangle."

The area bounded by this sort-of hypotenuse and a straight line between the two vertices forming this pseudo-hypotenuse (a line &radic;194 units in length and the true hypotenuse of a triangle with opposite sides 5 and 13), is a scalene triangle of area 0.5 square units.

The bottom "triangle," once you sum the area of the shaded regions and add the new square in, is 33 square units. This is due to the convexity of the illusory hypotenuse of the "triangle" that someone else mentioned earlier. Once again, the region bounded by a true hypotenuse and the apparent one is a scalene triangle of 0.5 square units.

Thus, the difference in area between the two figures 1 whole square unit.

Thus, the difference in area between the two figures 1 whole square unit.

OK.

I have asked for the maths explanation from the site (as I promised to post it by Sunday) by I do not think it will be much different from the explanation you posted.

Answer from the site:

Solution: The extra square results from the assumption that the hypotenuses of the two smaller triangles are in an exact straight line. The discrepancy amounts to about three percent of the large triangle's area.


So it is as most of you suggested.

I did not read every post, only the first 5-6, so maybe somebody already posted the following explanation.Both figures have a right triangle with legs, 2 & 5. Area is 5 units.

Both figures have a right triangle with legs 3 & 8. Area is 12 units.

One figure has a rectangle with sides 3 & 5. Area 15 units.

One figure has a rectangle with sides 2 & 8. Area 16 units.Add up the areas. One adds to 32, the other adds to 33. The two figures just cannot be the same. Forget how they look after a casual inspection of the diagrams. Many geometric fallacies are based on diagrams which are misleading.

As posted by others, careful inspection of the diagrams shows that they are not the same figure.

Dinosaur: your reasoning is correct. The difference between the two "triangles" lies solely in one square. (Some posters investigated more how the illusion of similarity arises (slight deviations in hypotenuse)

In the email from the site, the moderator says that the puzzle is simple to solve (where the differences reside) but what is surprising is that slight difference in of the hypotenuse results in <i> exactly</i> one square.

Well I agree with that.

Given by the Dutch site:

Solution

The extra square results from the assumption that the hypotenuses of the two smaller triangles are in an exact straight line. The discrepancy amounts to about three percent of the large triangle's area.

With a design program as PhotoShop, PhotoPaint, even Corel Draw, you can import the image and check for yourself.

See the original graphic:


<img src="http://mitosyfraudes.8k.com/images-6/puzzle03.gif">


And now and enlargred version of the hypotenuse critical point:


<img src="http://mitosyfraudes.8k.com/images-6/puzzle.gif">


There is another old puzzle like this, about a rectangle of gold that supposedly increses one square when arraged in a different fashion. An easy way of becoming rich.

When I posted my previous reply to this thread, I was tired. Instead of analyzing diagrams, one should do some elementary trigonometry.

Three right triangles are involved in this problem. If you take the ratio of the legs, you get the following. Atn(2/5) = 21.801 409 degrees. Atn(3/8) = 20.556 045 Atn(5/13) = 21.037 511

As many people don't "see" the solution just by looking at calculations (that gives the mathematical explanation for the "percieved" oddity), most people will "see" the explanation looking at a graphic as this: the extra square is equivalent to the area left out by the hypothenuse marked by the orange line.

<img src="http://mitosyfraudes.8k.com/images-6/puzzle2.gif">

Excelent explanation in graphics! It is surprizing that the difference in the angle translates precisely into on square of the required size. Is it an accident or is it possible to manipulate the whole think to get this outcome? My guess is it is really accidental: a graphic forming itselfs as a crystal would...

ProCop, if you watch closely the original graph, the presumed hypotenuse in the upper triangle is not a straight line, but is broken in two sections. The fist short one corresponds to the deep blue (small triangle) and the second section is the hypotenuse of the light blue triangle.

If you trace a right line between the two points of the big, big triangle, you'll get an area in the white part of the graphic, also equivalent to the area of the missing square.

<img src="http://mitosyfraudes.8k.com/images-6/puzzle04.gif">

Of course, the white area looks small because the orange line is too thick. But as Newton's First Law states, you cannot create anything out of nothing. And that should also apply to politician's promises... LOL. :D