I got this by email and have not spent much time on it for the sake of my sanity. So the question is please someone why o why 65 = 64? I don’t want to turn crazy! There must be a trick but I don’t see it!

It's an optical illusion. The pieces don't fit perfectly, and that's where the "extra" square comes from.

Patrick is righ, it is an optical illusion. Ignore the illusion and do this: Forget about the whole shape they make, it is an illusion, its not a proper fit. If you examine the edges you'll see mismatching. Calculate the areas of the geometric shapes individually in each case and you end up like this: Out of the orange and blue polygons, make the biggest rectangles you can and you are left with two smaller triangles. Now calculate the area for all the shapes and add them all up. It doesn't matter which shapes. Even if you calculate the area for the shapes in the new "false" shape, its still 64. It doesn't matter what you do. 1/2 base x height for the triangles and length x width for the rectangles for all the shapes = a total area of 64. The two big triangles=12 blocks each x 2 = 24 The two small triangles=5 blocks each x 2 = 10 The two rectangles are 15 each x 2 = 30 24 + 10 + 30 = 64 So there you go

Yup, it's not exactly 5 x 13, it's 4.944 x 12.944 or a similar slight deviation such that it still multiplies to 64.

I've seen a similar one done just with triangles. One way I found the problem with that one was to examine the slopes of the triangles. I found they weren't equal. Thus I was able to figure out the fact that the shapes weren't perfect triangles. I think the trick works partly based on the thickness of the grid lines and the thickness of the dividing lines. Those have to be taken into account, and when you look at this, you kind of look past those.

The diagonal line in the new shape is thinner at the upper right and thicker at the lower left. If you watch the shapes as they move around, the thickness of the surrouned lines change thickness.

This is a very old problem. Fun to see it come up again. Actually, there's a (very long and very thin) diamond shaped space in the center of the second image. It's a "hole" in the whole surface and its own surface is (you guessed it) exactly 1 square. So, because it's not part of the second shape, after subtracting that 65 - 1 = 64 again. Cees

Thats why if you ignore their trickery you can just assume that the lines they put on the shapes should go from point to point, meaning, from the corners of those squares and then do the math of the area geometry. Length x Width and 1/2 Base x Height will do you fine each time.

Ha! Makes me think of my old Calculus class in college where the professor would get a kick out of confusing freshmen by "proving" to them via some real convoluted formula that 1=0. He offered extra credit to the class to prove him wrong and no one could do it (he wrote the whole thing out on the chalkboard and I've since discarded all my notes from that class).

There is a book you folks should read. Its called: Zero. Biography of a Dangerous Idea. by Charles Seife Its neat. Most of you would find it interesting

Chris, I have not read that book but I have read "The Nothing That Is: A Natural History of Zero". Ironically I just looked it up at Amazon and noticed it was double-listed with Zero. Quite fascinating, I might add. I recommend it to everyone who's even slightly fascinated with math.