a puzzler, but tell me if you can figure it out =)..
Seems like the hypotenuse line on the first triangle is slightly concave
The hypotenuse line formed by the second triangle is slightly convex, and at the bulges' peak height, it frees up just enough space under the triangle so that one square is open. Put a ruler to it, and ignore the graphing lines.
Those aren't perfectly straight triangles.
I orginally typed this, which when I got to the drawing part I realized I
was wrong. Simply put without trying to prove it, the answer lies in the
rearrangement of the dark green and red portions.
"""The answer lies in the arrangement of the red and dark green portions. The red portion is a block higher than the green. So when placed on top of the orange and light green portions, it becoems one block too high. To solve this, they moved the orange portion down a block, which required it to be pulled over one block to fit which creates the one block (white) gap. But now if you followed me correctly, you will be asking yourself, then how come the entire triange isnt a block longer if it was pulled over a block? The answer to that question lies in the red and dark green portions. The answer lies in the difference in length of the base line of the red portion, and the top line of the dark green portion. Basically, when you switched positions with the red and dark green portions, you also switched lengths, by changing from the top ridge of the red block, to using the bottem ridge. This is a hard concept to explain, so here is a little geometry.
Lets name the red portion, Block A. Lets call the dark green portion Block B. The top ridge of Block A is called BA. The bottem portion, BB. The side portion, BS (haha)
Block A, BS length = 3 blocks long.
Block A, BA length = 8 blocks long
Black A, BB lenth = ?? 9 blocks long. How do we know? Our friend the pathagorean theorm.
BS = 3 squared = 9
BA = 8 squared = 64
Add them. BS + BA = 73
Now we take the square root of that number (73) which is the length of the hypotenuse squared. Then we will have the length of Block A, BB. Which is 8.54 Close enough, not exact because of the graph not being perfect or providing good guildance. So there you have it. """"
Oh well. What a waste. :ohcrap: I tried. I'm fried.
Whats got it and Dodgerida is almost there but needs a little more
information in his proof.
The basic explanation is that the slope (rise over run) of the red triangle is lower than the slope of the green triangle. This gives the covex and concave line shape What was talking about.
Now you may ask, so what? How does that explain it?
Well, if the 2 slopes were the same you would get a straight line from the upper right point of the entire triangle to the lower left point of the entire triangle. This is not the case.
If you were to draw a straight line from the upper right point to the lower left point of the upper triangle, you would see that there is some aread between the line and the upper edge of the triangle. If you did the same to the lower triangle you would see that the line actually lies inside the triangle and there is some area of the triangle above the line.
Now since the area is above the line in the lower triangle and below the line in the upper triangle, it has to shift from somewhere and that is why you get the extra square in the bottom.
Hope this helps some of you understand what is going on.
what got it right on the dot. dodge pretty much lost me there, but hes got it some what right. its actually an illusion... the triangles arent really the same size. thats all i can say :mrgreen:
There is no illusion about it. It is simple math. Each shape is the same size from top to bottom and the area covered by the shapes from top to bottom is the same. the difference is that there is a shift in where the area is applied and if you shift it to one area, you need to take it away from another. Thus the empty square at the bottom.
when i said "the trianlges arent the same size" i meant triangle A (one without the hole) and triangle B (one with the hole). just wanted to clear that up. the 'hypotenuse' does not equally match, its actually a quadrilateral.. i cant really explain it.. but it has little to do with the triangle as a whole, but rather the arrangement. its an illusion if you ask me, cause the slope is almost invisible to the naked eye.