Goldmund
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Mon Feb-07-05 03:58 PM
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BrklynLiberal
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Mon Feb-07-05 04:04 PM
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1. The red piece in the top and bottom are not of the same size.(area) |
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Edited on Mon Feb-07-05 04:10 PM by BrklynLiberal
The missing square is the difference in its area?
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Goldmund
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Mon Feb-07-05 04:06 PM
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2. How come then both fit together perfectly with other pieces? |
BrklynLiberal
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Mon Feb-07-05 04:13 PM
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7. Is there an answer to this? |
SheepyMcSheepster
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Mon Feb-07-05 04:10 PM
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4. the area seems the same to me? |
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:shrug:
how are they different? how can you tell?
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Worst Username Ever
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Mon Feb-07-05 04:13 PM
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Edited on Mon Feb-07-05 04:14 PM by Worst Username Ever
They are both 8 squares long and three squares high...
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bobbobbins
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Mon Feb-07-05 04:09 PM
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3. you sir, just blew my mind |
bobbobbins
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Mon Feb-07-05 04:11 PM
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5. im assuming the slope is slightly different on both triangles |
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resulting in the extra square being made up there somehow
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JVS
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Mon Feb-07-05 04:12 PM
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6. That shape is not really a triangle! |
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The rate of incline of the red triangle is 8/3 the rate of incline for the green triangle is 5/2. Thus the shape they form is not a triangle and misleads us about what the true area is.
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VioletLake
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Mon Feb-07-05 04:18 PM
Response to Reply #6 |
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Edited on Mon Feb-07-05 04:20 PM by VioletLake
Add: if you lay a straight edge over the top triangle, the top edge is slightly curved, and I guess that's where the missing space is.
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AllegroRondo
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Mon Feb-07-05 04:22 PM
Response to Reply #11 |
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Edited on Mon Feb-07-05 04:31 PM by AllegroRondo
The green triangle has a steeper slope than the red triangle.
In the first diagram, the green triangle is on top, so the long line formed by the green and red triangles bulges inward slightly. In the second diagram, the red triagle is on top, so the long line bulges outward slightly.
Its the shift in the bulge that makes the triangle as a whole get slightly larger in the second diagram.
look at the point where the red and green triangles meet in the first diagram, then find the same point on the other diagram. Its actually 'inside' the triangle on the second diagram.
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hfojvt
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Mon Feb-07-05 04:19 PM
Response to Reply #6 |
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but you may note that in the 2nd triangle, the slope is like 8 to 3.2 and thus comes your extra area. The tip of the red triangle has more area than the tail.
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Goldmund
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Mon Feb-07-05 04:22 PM
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14. That's the closest I can come to solving this too. |
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It's some sort of an optical illusion. The grid must not be totally uniform. If you try to calculate the height drawn from 8 units to the right of the left angle of the big 13x5 triangle, it won't exactly be 3. Or, if you try to calculate the left angle, you should be able to take arctan(5/13) and arctan(3/8) and come up with the same thing. But you don't -- the values are significantly different.
I can't exactly tell where and how the grid is not uniform, but that's the only possible solution to this.
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AllegroRondo
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Mon Feb-07-05 04:28 PM
Response to Reply #14 |
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print the page and cut out all the pieces. Place the green triangle on top of the red one, and compare their slopes.
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Goldmund
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Mon Feb-07-05 04:29 PM
Response to Reply #19 |
20. The difference is so slight that this is unnoticeable. |
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But when it gets squared, it produces enough surface area discrepancy for one squared unit.
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On the Road
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Mon Feb-07-05 04:27 PM
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I didn't notice it until you pointed it out. All the individual pieces seem to match.
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hfojvt
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Mon Feb-07-05 04:15 PM
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9. If you look at the 6th, 7th, 8th columns from the left |
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you will see that in the bottom diagram they are taller that is where the area goes to produce that hole because the red area in columns 6,7,and 8 in the first diagram is replaced by the larger red area in columns 1,2,and 3
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aden_nak
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Mon Feb-07-05 04:16 PM
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10. All the pieces are the exact same size. I measured them in Photoshop. |
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Edited on Mon Feb-07-05 04:18 PM by aden_nak
The only thing I can think of is that it has something to do with the perimeters. That because the perimeters that touch are different, you are making up one square's worth of extra space on the black lines.
Eh?
EDIT: They also travel along the exact same slope. Measured that, too.
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Goldmund
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Mon Feb-07-05 04:23 PM
Response to Reply #10 |
17. You measured it pixel-by-pixel? For real? |
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Edited on Mon Feb-07-05 04:55 PM by Goldmund
On edit: yes, that doesn't surprise me -- they are the same. But the compound shape isn't exactly a triangle, as several posters noted.
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BrklynLiberal
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Mon Feb-07-05 04:20 PM
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13. It is a cool puzzle! Anyone know a math teacher? |
Goldmund
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Mon Feb-07-05 04:23 PM
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16. The only thing that math will help you do... |
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...is figure out that this isn't exactly a math problem. It's an optical one.
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unblock
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Mon Feb-07-05 04:43 PM
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21. first, stop thinking of the compositions as triangles, because they're not |
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they've both got FOUR sides, not three. what appears to be the hypotenuse of a big triangle is actually two lines that aren't quite at the same angle. so the two composites are actually quadrilaterals, and the top one is slightly concave while the bottom one is slightly convex.
turn your monitor and look down the length of the shapes to see.
the difference in area happens to be equal to one square unit, which makes room for the gap in the bottom, convex composite.
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maveric
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Mon Feb-07-05 04:58 PM
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22. I printed it out and put a precision protractor on both triangles. |
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The top one is @ 21 degress, the bottom @ 22 degrees. The grids may not be the same size. I should check the length of the hypotenuse.
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JVS
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Tue Feb-08-05 04:35 PM
Response to Reply #22 |
23. Just count the length of triangles by using the grid boxes |
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Wed May 08th 2024, 04:22 AM
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