It seems that the four shapes on the first figure have the same area as the other four shapes. On the other hand, obviously the area of the rectangle is 65 while of the square is 64.

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One square is lost because the square space has even sides and the rectangular space has odd sides. Multiplying two even sides of a square equals an even number versus multiplying the odd sides of the rectangle equals an odd number. The different areas that surround the four shapes is the trick, not the shapes themselves which are all equal. I think?

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The red and yellow shapes keep the same area between the left and right figures. A white square space was lost when rearranging from the left figure to the right figure. The white area above the red shape on the left has an area of 5 and the white area above the yellow triangle has an area of 28 for a total of 33 white squares. In the right figure, the white space below the red shape has an area of 20, and the white space below the yellow triangle has an area of 12 for a total of 32 white squares.

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...yes. i.e. the diagonal line on the left figure actually is not a line but quadrangle – with area 1. So, the vertices of this quadrangle are: top left, point which seems to be on the diagonal where red and yellow shapes border, point which seems to