Take a 12x9 rectangle and cut an 8x1 hole in the middle, with the hole centered exactly in the rectangle (I think of 12 and 8 as being the horizontal dimensions). Can you make two cuts so that the resulting pieces can be rearranged to form a square?

The knife used to make the cuts is not restricted to cutting along a straight line: it can turn as it cuts.

The Mathematica code for the diagram is attached below.

Source: G. Polya: Mathematical Discovery

Show[Graphics[{GrayLevel[0.5], Rectangle[{0, 0}, {12, 9}], GrayLevel[1], Rectangle[{2, 4.5}, {10, 5.5}], GrayLevel[0], Line[{{0, 0}, {12, 0}, {12, 9}, {0, 9}, {0, 0}}], Line[{{2, 4.5}, {10, 4.5}, {10, 5.5}, {2, 5.5}, {2,4.5}}], Text[12, {6, -0.6}], Text[9, {-0.4, 5}], Text[1, {1.7, 4.8}], Text[8, {6, 3.9}]}], TextStyle -> {FontSize -> 12, FontWeight -> Bold, FontFamily -> "Times"}];© Copyright 1998 Stan Wagon. Reproduced with permission.

5 October 1998