A mean little problem

[Goldmund]

[BrklynLiberal]

1. The red piece in the top and bottom are not of the same size.(area)

The missing square is the difference in its area?

[Goldmund]

2. How come then both fit together perfectly with other pieces?

[BrklynLiberal]

7. Is there an answer to this?

:shrug:

[SheepyMcSheepster]

4. the area seems the same to me?

:shrug:

how are they different? how can you tell?

[Worst Username Ever]

8. How can you tell?

They are both 8 squares long and three squares high...

[bobbobbins]

3. you sir, just blew my mind

[bobbobbins]

5. im assuming the slope is slightly different on both triangles

resulting in the extra square being made up there somehow

[JVS]

6. That shape is not really a triangle!

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.

[VioletLake]

11. I think you got it.

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.

[AllegroRondo]

15. You got it

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.

[hfojvt]

12. I believe it is

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.

[Goldmund]

14. That's the closest I can come to solving this too.

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.

[AllegroRondo]

19. Try this for help

print the page and cut out all the pieces. Place the green triangle on top of the red one, and compare their slopes.

[Goldmund]

20. The difference is so slight that this is unnoticeable.

But when it gets squared, it produces enough surface area discrepancy for one squared unit.

[On the Road]

18. That's It

I didn't notice it until you pointed it out. All the individual pieces seem to match.

[hfojvt]

9. If you look at the 6th, 7th, 8th columns from the left

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

[aden_nak]

10. All the pieces are the exact same size. I measured them in Photoshop.

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.

[Goldmund]

17. You measured it pixel-by-pixel? For real?

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.

[BrklynLiberal]

13. It is a cool puzzle! Anyone know a math teacher?

[Goldmund]

16. The only thing that math will help you do...

...is figure out that this isn't exactly a math problem. It's an optical one.

[unblock]

21. first, stop thinking of the compositions as triangles, because they're not

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.

[maveric]

22. I printed it out and put a precision protractor on both triangles.

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.

[JVS]

23. Just count the length of triangles by using the grid boxes