how is this possible? I know there is some trick, should someone please explain?!

6This appeared here in a similar post using triangles rather than squares. The lesson here is that drawings are not proofs. – Asaf Karagila Oct 27 '13 at 12:00

What program do you need to create such animations? I'd enjoy to make some animated recreational math problems... – Jeyekomon Oct 27 '13 at 12:06

2Note: 面積 = Area, 甲/乙/丙/丁 are labels. one should interpret them as A / B / C / D or $1^{st} / 2^{nd} / 3^{rd} / 4^{th}$ in this question. – achille hui Oct 27 '13 at 12:06

http://www.cuttheknot.org/Curriculum/Fallacies/FibonacciCheat.shtml or http://brainden.com/forum/index.php/topic/1396465geometryparadox/ – lab bhattacharjee Oct 27 '13 at 14:39

In this particular drawing of the result, the precision of the drawing is such that I think you can actually see the missing area as a thickening of the line along the diagonal of the rectangle. – David K Jul 12 '21 at 12:27
2 Answers
This is a very well known optical illusion. Count the number of squares in each triangle (or at least in each nonvertical or nonhorizontal line) and you'll see that they don't have the same slope. Therefore the triangles cannot magically ''fit'' as they seem to do so.
The slope of green and red is 3/8 (0.375), where as the slope of blue and orange is 2/5 (0.4). These numbers are quite close so it's easy to hide one square unit. But the slopes cannot fit the way they look like they do.
Hope that helps,
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The gradient of the green triangle is not the same as the blue quadrilateral, this creates the overlap. Try to calculate the gradient (rise over run) of each sloping side yourself. Since it isn't equal, there is some overlap in the second figure, thus the "extra" square is hidden in the small overlapping sliver.
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