ARRRGGH!!! Need help with a puzzle!

[dr_mabeuse]

I’m going crazy over a puzzle and I need help. Unfortunately, I can’t paste the puzzle here, all I can do is give a link to the page where it appears. Here’s the link:

http://www.original-cards.com/

Go down to the third row from the top, where it says “Illusions” and open that. It’ll bring up a bunch of optical illusions in postcard formate. The puzzle I need help with is in the second row down, fourth from the left: the two triangles.

It shows two triangles composed of smaller geometric forms. The geometric forms are the same in both triangles, only rearranged, and yet in the second triangle, as if by magic, an extra square of area appears.

The overall area of the triangles are the same. (For those of you who don’t remember, the area of a triangle is given by ½ base X height) The area of the internal shapes is the same. Where the $*^!* does the extra square of area come from?

I’ve been thinking about this and turning it over in my head for a month or two now, and I need help. If someone can explain to me how this can be, I’d be most thankful.

---dr.M.

BTW, if you don't know anything about geometry, this might not mean anything to you at all. And if you’re the obessive type, maybe you should just ignore this.

 

[The_Fool]

Shit........

 

[Guest]

Shit........

LOL! That was my first reaction.

Doc, I wish I'd heeded your advice about the obsessive types ignoring this.

 

[Guest]

I stared at that one for a couple hours a few months ago. All I can say is that it's a mathamatic/geometric anomoly.

 

[The_Fool]

Area of Triangle (red) = [0.5(8*3)] = 12
Area of Triangle (Blue) = [0.5(5*2)] = 5
Area of Region (Orange) = 7
Area of Region (Green) = 8
Total Area = 32

Area of Total Triangle = [0.5(13*5)] = 32.5

Shit don’t add up….

 

[CharleyH]

LOL Merry Christmas Doc!!! Let's get Lauren to do it

 

[Guest]

Area of Triangle (red) = [0.5(8*3)] = 12
Area of Triangle (Blue) = [0.5(5*2)] = 5
Area of Region (Orange) = 7
Area of Region (Green) = 8
Total Area = 32

Area of Total Triangle = [0.5(13*5)] = 32.5

Shit don’t add up….

That could help explain it. In the top configuration, the shapes fit within the area of the large triangle, in effect "rounding down" the 0.5 excess.

In the bottom configuration, they don't fit, so "round up" the 0.5 excess, making it a whole 1 extra square.

It's the way the orange and green slot together in the top one, but don't in the bottom.

I'm clueless, really!

 

[CharleyH]

Ah, Lauren got it in 5

 

[Guest]

Ah, Lauren got it in 5

Thought she might! Come on, Lauren, spill it!

 

[Lauren Hynde]

The second "triangle" isn't a triangle. The hypotenuse isn't a straight line, it's slightly convex. It's very slight, but you can see it by the way the grid intercepts each of the triangles; that's where the extra area comes from.

(Took me about 2 seconds to figure it out. I guess studying architecture does pay off after all... )

 

[Evil Alpaca]

That could help explain it. In the top configuration, the shapes fit within the area of the large triangle, in effect "rounding down" the 0.5 excess.

In the bottom configuration, they don't fit, so "round up" the 0.5 excess, making it a whole 1 extra square.

It's the way the orange and green slot together in the top one, but don't in the bottom.

I'm clueless, really!

But I keep thinking there should be some kind of overlap . . . I wish I had some physical representation rather than pictures on the internet. It's easier to think about it when you can actually move the pieces.

ARGH!

 

[Colleen Thomas]

I’m going crazy over a puzzle and I need help. Unfortunately, I can’t paste the puzzle here, all I can do is give a link to the page where it appears. Here’s the link:

http://www.original-cards.com/

Go down to the third row from the top, where it says “Illusions” and open that. It’ll bring up a bunch of optical illusions in postcard formate. The puzzle I need help with is in the second row down, fourth from the left: the two triangles.

It shows two triangles composed of smaller geometric forms. The geometric forms are the same in both triangles, only rearranged, and yet in the second triangle, as if by magic, an extra square of area appears.

The overall area of the triangles are the same. (For those of you who don’t remember, the area of a triangle is given by ½ base X height) The area of the internal shapes is the same. Where the $*^!* does the extra square of area come from?

I’ve been thinking about this and turning it over in my head for a month or two now, and I need help. If someone can explain to me how this can be, I’d be most thankful.

---dr.M.

BTW, if you don't know anything about geometry, this might not mean anything to you at all. And if you’re the obessive type, maybe you should just ignore this.

The space is still part of the triangle Doc. If it isn't included then the figure isn't a triangle. The entire area coverd is the same, if you include the space. If you exclude it, are no longer matters since it isn't a triangle.

 

[Guest]

The second "triangle" isn't a triangle. The hypotenuse isn't a straight line, it's slightly concave. It's very slight, but you can see it by the way the grid intercepts each of the triangles; that's where the extra area comes from.

(Took me about 2 seconds to figure it out. I guess studying architecture does pay off after all... )

Oh yes!

But, don't you mean convex?

I was thinking it must be an illusion (seeing as that's what the puzzle's category is. LOL!)

You da woman!

 

[Lauren Hynde]

Oh yes!

But, don't you mean convex?

I was thinking it must be an illusion (seeing as that's what the puzzle's category is. LOL!)

You da woman!

Yeah. That one.

 

[nikkijames]

I'm sure that others will come up with this also.. but

This is because that although the hypotenuse of the large composite triangle looks as if it's a straight line, it's not.
From my school days and SOH CAH TOA (Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse and Tan = Opposite / Adjacent, we find that for the smaller blue triangle the bottom angle is 21.8, whilst for the bigger red triangle it is 20.56, This small difference (1.24 degrees) is enough so that in the first composite triangle the "hypotenuse" is actually concave and in the second convex, giving the extra unit of space.

 

[Evil Alpaca]

The second "triangle" isn't a triangle. The hypotenuse isn't a straight line, it's slightly concave. It's very slight, but you can see it by the way the grid intercepts each of the triangles; that's where the extra area comes from.

(Took me about 2 seconds to figure it out. I guess studying architecture does pay off after all... )

Is there really that much extra space to be gained that way? I saw a little discrepency, but I thought it was just an afteraffect of the medium they're displayed on.

Double-ARGH!

 

[Guest]

Yeah. That one.

PMSL!!! I could hear you say that.

 

[Guest]

I'm sure that others will come up with this also.. but

This is because that although the hypotenuse of the large composite triangle looks as if it's a straight line, it's not.
From my school days and SOH CAH TOA (Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse and Tan = Opposite / Adjacent, we find that for the smaller blue triangle the bottom angle is 21.8, whilst for the bigger red triangle it is 20.56, This small difference (1.24 degrees) is enough so that in the first composite triangle the "hypotenuse" is actually concave and in the second convex, giving the extra unit of space.

Yay! Nikki got it, too!

 

[CharleyH]

Let's get Lauren to post more. She turns me on almost as Sub Joe Ya well . . .

 

[dr_mabeuse]

Awful Arthur just PM'd me and pointed out something critical. Triangles are called 'similar' when they have the same configuration of internal angles. If triangles are similar, then their sides should be proportionate. The two internal trainagles appear to be similar, but they're not.

The ratio of the two short sides of the small triangle is 2:5. In the larger triangle they're 3:8. Expressing both these ratios in their least common denominator gives the small triangle the ratio of 16:40, while the larger triangle is 15:40. Therefore, they're not really similar triangles.

Is this enough to account for the missing space? I have no idea. AA pointed out that the hypotenuse of the big triangle isn't as straight as it looks. He might be right. Those lines do seem suspiciously thick...

---dr.M.

 

[Lauren Hynde]

Is there really that much extra space to be gained that way? I saw a little discrepency, but I thought it was just an afteraffect of the medium they're displayed on.

Double-ARGH!

The little discrepancy looks to me almost one fifth of the height of a square and over a third its width, and it's spread out across a huge hipotenuse. I could calculate the deviation, but there's no point. The result would be one whole square. Trust me.

 

[Evil Alpaca]

The little discrepancy looks to me almost one fifth of the height of a square and over a third its width, and it's spread out across a huge hipotenuse. I could calculate the deviation, but there's no point. The result would be one whole square. Trust me.

I'll take your word on it. I bow and grovel before your geometric greatness. (Don't tell Vella . . . she might get mad that I groveled to someone else).
Again, I wonder if this illusion would've worked if we had a physical manifestation of the pieces rather than a drawing.

We may never know . . .

(Twilight Zone musics cued in the background)
(Fade to black)

 

[Lauren Hynde]

I'm sure that others will come up with this also.. but

This is because that although the hypotenuse of the large composite triangle looks as if it's a straight line, it's not.
From my school days and SOH CAH TOA (Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse and Tan = Opposite / Adjacent, we find that for the smaller blue triangle the bottom angle is 21.8, whilst for the bigger red triangle it is 20.56, This small difference (1.24 degrees) is enough so that in the first composite triangle the "hypotenuse" is actually concave and in the second convex, giving the extra unit of space.

Yes! I think it's actually easier to see on the thumbnails page, with the triangles scaled down.

http://cards.original-cards.com/illusions/thumbs/driehoeken.jpg

 

[Guest]

I'll take your word on it. I bow and grovel before your geometric greatness. (Don't tell Vella . . . she might get mad that I groveled to someone else).
Again, I wonder if this illusion would've worked if we had a physical manifestation of the pieces rather than a drawing.

We may never know . . .

(Twilight Zone musics cued in the background)
(Fade to black)

Nope, it would never have fooled us, because to re-create the two large "triangles" we would've needed four of the smaller triangles (two of each colour). They are different shapes. It works no other way than as a graphic. It is very much an optical illusion.

 

[Liar]

I've seen this one before, and had thwe solution explained to me, so I didn't want to spoil the fun by butting in too soon.

Lauren is right . One of those two (the one with the extra square, I guess) is in fact not a triangle. It only looks that way because it's almost one, and since we're told that it is one, we believe it. And if our eye-sight raises objections, our brain usually just tells it to STFU.

#L