How Come?

The trick is that the green triangle and the red triangle are not similar. So the two different configurations above are actually triangles with different areas, which is why the blank square appears. The reason the two triangles have different areas is because they have different angles because as I said above, the red and green triangle are not similar, so the angle at the apex when the green triangle is there is not the same angle at the apex when the red triangle is there.

To see this, start from the point on the left for each triangle and go five full squares. Now go up. The top triangle is less than two full squares in height at this distance, but the bottom triangle is exactly two full squares in height at this distance. So you see, although they look the same, they are actually different triangles.

Roger. This just the sort of tendentious reasoning that I'd expect from someone who's trying to cover up the worldwide Cosmic ray/Pyramid conspiracy.

I can see from this post that you are one of the cognoscenti. Therefore, I will entrust to you a few details on an over-unity generator Europium and I worked on not too long ago. It is based on precisely the geometry of this puzzle. I would post details right here, but am concerned that perhaps those details could fall into the wrong hands (Roger's for example). Therefore, I am instead providing this link, with the warning to the non-cognoscenti that following this link will cause your teeth to fall out, or worse.

Another way of saying what Roger just said:

1. If you look very closely, and use a straightedge, you can see that the "hypotenuse" in the upper picture is concave, and is convex in the lower. Therefore the "hypotenuse" in not really a hypotenuse: it's two sides of a quadrilateral.

2. You can check the slope of the red triangle hypotenuse and the green triangle hypotenuse. 3/8 = .375 2/5 = .4 The upper quadrilateral is therefore larger that the triangle it appears to be, and the lower quadrilateral is smaller than triangle it appears to be. The difference in size is the one unit square "hole."

A while ago, I cut out paper pieces like this and gave them to my kids to fool around with. Even with the pieces in hand the subterfuge is not obvious. A version in wood would make a good coffee table or desk conversation starter.

Thanks Ken. I was using "similar" in the geometric way, which is why I imbedded the link to its definition when I wrote it in that original post.

Somehow I missed that they are quadrilaterals, which I think is the most intuitive answer, so thanks for the explanation.

By making re-arrangements of 1st rectangle as shown in the 2nd fig. we can't get perfect triangle.The whole shape does not give tri-angle thats is the fact.

the hypotenuse line of 1st fig. is drawm between the 2 points and it is perfect tri-angle.

but in the 2nd fig it is the connection of two tri-angles such as red and green and this is not perfect straight line. thats y blank space.

but the area of shaded part in the 2nd fig same that of 1st one.therefore 2nd is not triangle.hence u cant make Q why blank?

the hypotenuse line of 1st fig. is drawm between the 2 points and it is perfect tri-angle.

It may appear so, but it is not. The "hypotenuse" in the first figure "sags" by the same amount that the "hypotenuse" of the second figure is nudged upward. The best way to verify this is to calculate the slopes of the hypotenuses in the red and green triangles: they are different (.375 vs .4). Therefore, neither the first figure nor the second can be a triangle -- they are both quadrilaterals.

Or, if you don't have straight edge, or don't want to put one against screen, note where the apexes of the little triangles and hypotenuse (after they moved) do not intersect the grid intersections the same from upper to lower figure, indicating that the wannabe hypotenuses are not following the same paths, upper and lower.

I meant to say, wannabe congruences. If the hypotenuse of one little triangle was concguent to that of the other, then at both the upper and lower big quad the seeming but not actual big hypotenuse would intersect intersections in the grid at the same points, upper figure and lower figure. They do not.