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Milan Fiori
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 Posted: Sun Feb 25, 2007 6:19 am Post subject: Math Teaser |
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Ok, we're having a bit of a heated debate in Zexen, which needs to be solved. I know there's lots of mathematically gifted people here at the site. So maybe you can help.
http://www.justriddlesandmore.com/wedge.gif
Now the whole point is to try and explain the gap on the bottom triangle mathematically. Which I can't.
I say, the gap is there, simply because the orange piece was moved so that it doesn't fit evenly with green piece anymore. I don't know if that even can be explained mathematically, but that's my answer.
However. Our righteous leader begs to differ with me. He says, that because the exact same shapes are used, the area of the triangle has to be the same. No matter what. So what's his answer? It's that the bottom triangle as a whole is slightly bent. Which makes absolutely no sense to me. Since it's clearly stated that both triangles are made up of the same things. So if the bottom is bent, so is the top, so that point is moot.
But like I said, it's getting pretty heated because he's adamant about being right. Where I'm adamant of him seeing things. So maybe someone here can shed some light on our problem!
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Hawk Thanatos
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| Posted: Sun Feb 25, 2007 8:00 am Post subject: |
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They aren't actually the same shape. If they were they'd take up the same space wouldn't they? The slope of the "hypotenuse" is different in both "triangles" (they aren't actually triangles). Here:
For this to have worked, the red triangle and the green triangle should have the same ratio. The green one is 5 across and 2 up (5/2=2.5), while the red one is 8 across and 3 up (8/3=2.67). That's why if you look at the two spots that I've circled you can see that the bottom "triangle's" "hypotenuse" is slightly higher at that point than the top one. And the space that it being slightly higher takes up is compensated with the empty space. |
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Ikano
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| Posted: Sun Feb 25, 2007 12:24 pm Post subject: |
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To put it simply...This puzzle is an illussion
According to geometric rules if the red and green triangles are similar (and they'd have to be for them to be able to fit into a perfect Triangle)
The ratio of the base of one triangle over the base of the other has to equal the ratio of the non-hypothenuse side of the first triangle over the non-hypothenuse side of the other.
8/5 has to equal 3/2
However 8/5 = 1.6 and 3/2 is equal to 1.5
You can further see there is something odd with the areas in the first shape if you treat the first figure as a triangle and calculate the individual shapes areas within the triangle:
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total area: (13 x 5)/2 = 32.5
red triangle: (3x8)/2 = 12
green triangle: (2x5)/2 = 5
total area minus the triangles: 15.5
area of the remaining shapes = 15
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There is a missing .5 Where did it go? It could not possibly be a perfect triangle without that missing .5
Lastly.....You can try taking a piece of paper and hold up one of it's straight edges to the hypothenus side of the large triangles.
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shuxion
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| Posted: Sat Mar 10, 2007 11:29 pm Post subject: |
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Kind of cool. The took the larger and shorter parts and moved them to opposite sides. That makes the space with the triangle being whole. That is the best i can explain it. I am horrible at math but i do see the correlation. Have any more???
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eXistence of Fly
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| Posted: Sun Mar 11, 2007 12:10 am Post subject: |
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| Hawk Thanatos wrote: |
| They aren't actually the same shape. If they were they'd take up the same space wouldn't they? The slope of the "hypotenuse" is different in both "triangles" (they aren't actually triangles). |
The slope looks to be an error in 'printing' as for that gap to be caused by the space between gaps as you higlighted they'd have to be more than about 1/10th the space of a square. With that directly contradicting the statement that all parameters of the shapes were unchanged in Figure 2 we can assume that small gap is an error in 'printing'
With right angled triangles of these varieties it is possible to line them up so that their slopes match evenly if supported. Since in the first diagram the green triangle is supported by the 'tetris' blocks to line up with the height of the red triangle, and this is likewise the case in the 2nd picture.
Thus this isn't a mathmatic problem, its a geometric problem cause by the 'tetris' shapes.
Since the first and second pic the giant triangle (the one which comprises all 4 parts) is kept the same the gap is there due to the shifting of the orange and light green shapes to fill the required height for the dark green triangle to align with the red triangle. Since both blocks are mirrored of each other (5x3, 5x2) there is no way unless you flip them vertical that they will lock in together as seen in figure 1. They can only form a rectangle of 5x3.
Instead what they tried to do was create a rectangle of 8x2 which brings about a 16 square rectangle with 1 gap overlay due to the 3 extension attempting to overlap the 2 extension.
- - - - - - - -
- - - - - - - -
As you can see, in order to line up those two shapes in the order that you see in figure 2 there is a gap as denoted by the pink -.
So theres no mathmatics required, just simple geometry.
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Milan Fiori
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| Posted: Sun Mar 11, 2007 12:38 am Post subject: |
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Grink?
My original answer was the short version of Fly's answer. He's just more eloquent than I am. However Tonberry's answer was what Hawk was saying.
So uh... which answer is the right one?
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eXistence of Fly
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| Posted: Sun Mar 11, 2007 12:54 am Post subject: |
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| Milan Fiori wrote: |
Grink?
My original answer was the short version of Fly's answer. He's just more eloquent than I am. However Tonberry's answer was what Hawk was saying.
So uh... which answer is the right one? |
Mine of course. :P
Right angled triangles share the same base angle so that the only thing which changes with the triangles size is the length of the slope and not the angle of it. The picture has 3 right angled triangles:
The red one
The dark green one
The big one
In Figure 1 the dark green triangle is encompassing the orange and green blocks to creatke half of the giant triangle, since figure 2 the gaps hawk pointed out are still there upon the green triangle (4 across 2 up from the bottom left, 6 across 3 up from the bottom left) to match with Figure 1s section the amount of loss incurred by the slope is the same.
Right angle triangles are easy to line up and no gap perpetrated by a slight difference in their slope can account for tthe entire gap in figure 2.
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Sierra Mikain
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| Posted: Sun Mar 11, 2007 1:06 am Post subject: |
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Neither of the finished figures are triangles. They both have 4 sides and one right angle. What appears to be a hypotenuse because you are ASSUMING it is a triangle, is indeed 2 different slopes and not 1.
While they both have the same slope for the same amount of time, the average width of the figure has changed.
Take this figure I just drew. The slopes are more different, but the average slope in the figures are the same with the 2 slopes simply swapped. With the slopes more drastically different, it's easier to see.
In short, with 2 different shapes, it makes no sense to assume they both have the same area. The average width of the figure has changed and is compensated by an empty space. This is what hawk said. They are not the same shape. Hawk is right.
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Ikano
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| Posted: Sun Mar 11, 2007 1:47 am Post subject: |
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| Sierra Mikain wrote: |
| Hawk is right. |
Hey I was right too
....I just...explained it differently...
eheh...erhm...*cough*
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