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Old mathematical fallacy done with some clever geometry. You can prove any number is equal to any other number with some basic math. The explanation is available on the web and has been discussed before on HAN.
I didn't read any of those, but it involves taking a slight amount out of each square to add another in the end, I'll show you a similar animation that uses the same principle.
its because when you cut it up and reaarrange it to other rectangle, you get the same 64 squares, but 1 extra "square" from the original 8x8 square perimeter.