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These kinds of puzzles depend on tricking the eye into thinking things that are not straight lines are straight, or that things that are not squares are squares. Look at the video closely, and you can see some not-quite-square bits.

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See the jump-cut at 0:37? The reason for that is that he sneakily re-arranged the upper pieces in the tray, and added the 3 squares that he later removes. When he lays everything out on the table at first, it doesn't fit very well at the diagonal. It is not the same arrangement of pieces that was in the wooden frame originally.

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