a simple maths question!

That's pretty cool. I believe that the squares are being added together in a different way than they are being divided up, and that results in the addiitional square. But I don't really know.

 

Piece of cake, as it's an optical illusion, powered by the outlines on the sections they move. First, they make two slices and rearrange things a bit. The blue slice creates two triangles, and the red slice creates two trapezoids. Rearrange as desired, and without any outlines to cover things up, this is what you get:

See that white gap? It's a parallelogram with an area of 1, which is why 64 seemingly equals 65. Warning: geometry and trig follow. Start off noting that the two blue triangles are identical, as are the two red trapezoids. For the blue triangle: looking at the smallest angle, you know the adjacent side is 8, and the opposite side is 3.

32 + 82 = x2, so the hypotenuse is 8.544 tan-1(3/8) = 20.556° For the red trapezoid: look at the slice that happened. Basically, it took a triangle (green) out of a square to make the red trapezoid.
On that triangular piece, looking at the smallest angle, you know the adjacent side is 5, and the opposite side is 2. 22 + 52 = y2, so the hypotenuse is 5.385 tan-1(2/5) = 21.80°

Given that it's the slice that was removed from the 5 x 5 square to result in the red trapezoid, you know the acute angle of the trapezoid and the length of the slanty side. 90° - 21.80° = 68.20°
Length of slanty side = removed section hypotenuse = 5.385

Now you know the acute angle of the parallelogram in the middle. 90° - 68.2° - 20.556° = 1.244°. So now you know everything you need to find the area of the parallelogram: opposite sides blue are 8.544, opposite sides red are 5.385, and the acute angles are 1.244°.

Area of a parallelogram is base times height, so you need to draw a right triangle inside. The following diagram is totally skewed for clarity, but the values are accurate.

See how easy this is!

 

Its an illusion. When u do it precisely, there is a thin space between the diagonal line going left to right.

 

That's what I said, but explained the details of the math.

 

ooops, i didn't see ur post =)

 

Of course you can't do it. All lawyers can add up is hours!

 

it's not an illusion, I think it has something to do with the slopes of the lines.. not sure how to explain, maybe someone else could

 

Did you not read post #3?

 

a very neat visual teaser.
thanks for the explanations, Peter.

 

check my profile for a similar one.

 

Yours is interesting as well. I see that the hypotenuse in the second arrangement describes an area which would be equal to the 1 box square left blank.

 

WTF???--Peter you lost me after piece of cake--I started to get hungry and lost all concentration..