Brain Teaser of the Day...and, no, I don't know the answer.(Maybe you can help)

[Dallas59]

HA HA HA! All your holes belong to us!

[Darkwolf]

I don't know good math talkin' but doesn't the red triangle's longer base make the "back" end of the triangle bigger in this arrangement, while not affecting the overall length? (Imagine a long car towing a small trailer--if you switch positions, the "long" part is in back now.)

[HairOfTheDog]

~scratches head~

[jdm]

In the top triangle, the green piece is below the yellow piece.

In the bottom triangle, the green piece is next to the yellow piece.

Such a change in placement would account for the extra square on the grid.

[MikefromOhio]

Count the squares the bottom of the red triangle takes versus the number of squares the bottom of the green triangle take up....

[HairOfTheDog]

But the sum of the shapes should have the same overall area regardless of how you arrange them.

[Mr. K]

it is hard to see, but triangle 2 has more area enclosed

[MikefromOhio]

not according to the orange block :)

how do I explain this one...

ummm......crap....

[YankeeinOkieville]

8:3 5:2

[Swordmaker]

A Two unit high by five unit long right triangle does not have the same angle at the top angle as a three unit high by eight unit long right angle. The first set up has a subtle but noticeable downward bend along the top side where the two triangles meet. When you reposition the pieces, the bend is reversed, bending upwards, adding area to the over all triangle. This differential is equal to one square unit.

[HairOfTheDog]

There is an optical illusion that would suggest an arc, but there is no arc. I confirm with Photoshop that the hypotenuse is a straight line in both.

[Graybeard58]

The shapes in the second figure are no longer interlocking. the interesting thing is that there is only one hole, because you can change the figures in any number of ways and create even more holes but the pieces would still be contiguous.

[ecurbh]

OK, there is something hinky going on here...

Area = Base x Height / 2

Overall area of both shapes should be 32.5 (13 x 5 / 2)

Area of the red triangle is 12 (8 x 3 / 2)

Area of the green triangle is 5 (5 x 2 / 2)

Area of the other shapes are 7 and 8...

But 12 + 5 + 7 + 8 = 32!

That missing .5 is the key to this. But that's where I get stuck. My brother is a mathematics professor, I'll have to email this to him.

[dinasour]

The hypotenuse of the top triangle is not a straight line, but instead the red and green triangles make a small angle where they meet.

[Darkwolf]

"But the sum of the shapes should have the same overall area regardless of how you arrange them."

Of course. And...?

[dinasour]

Actually, the top one is bowed in, and the bottom is bowed out.

As noted above, 5:2 != 8:3 which means that the red and green triangles are not similar, therefore have different internal angles.

[KC_for_Freedom]

Yes, two different slopes, therefore not a straight line.